advanced-pdes

[sinful herald]

I am reading a book on functional calculus, where I have doubt in definition, can someone help me understand (1.3), I do not understand why it is well defined since $\phi(ef)$ might or might not be in domain of $\phi(e)^{-1}$

[untold delta]

Math&Tea

[quick pagoda]

What exactly does the L(X) stand for here

[sinful herald]

Linear operators

[quick pagoda]

Any continuity or closedness or domain constraints

But anyway, the domain of phi(f) may not be the whole space

Consider applying 1 to, say, the laplacian

On L^2

It’s got a dense domain but not entire

[sinful herald]

yes, I have doubt in the definition, I dont see why this make sense

[quick pagoda]

You seem early on in the text ofc so applying to a particular setting might be a tad forward

But point is, if f is only regularizeable, it need not be defined everywhere on x

[sinful herald]

yes that I agree

but how is that a problem?

[quick pagoda]

And, in fact, won’t be

It isn’t

Unless you want it defined everywhere, of course

Then you might be in trouble

[sinful herald]

ohh so you are saying only take those elements for which phi(ef) lies in domain

and make that domain of phi(f)

[quick pagoda]

Ye

Now, I’m sure they cover that soon, but this depends on e, a priori

So why is it safe to pick an e

[sinful herald]

it has to be show that definition is independent of e

[quick pagoda]

Exactly

Especially a bit tricky for domain issues though, but they probably have that soon in the text

[sinful herald]

ok, I see the domain thing now. I didn't even though of that before you mentioned. Thanks a lot

[quick pagoda]

Unbounded operators are always tricky, but note that \Phi is gonna “”look like”” applying the “functions” e to a specific operator (namely, \Phi(1))

[rotund jetty]

Does anyone know bounds for $\sum_{j, k = 0}^{T - 1}Var(f_j f_k)$, where $f_0, f_1, \dots$ are eigenfunctions of $-\Delta$ on a compact manifold $M$ without boundary, normalized so that $E(f_j^2) = 1$? This paper, https://arxiv.org/pdf/1011.0215, bounds $\frac{1}{T}\sum_{j = 0}^{T - 1}E(f_j^4)$ on $S^2$, which in turn gives a bound for $\sum_{j, k = 0}^{T - 1}Var(f_j f_k)$ that is quite good. Are there papers that bound $\frac{1}{T}\sum_{j = 0}^{T - 1}E(f_j^4)$ for general $M$?

[untold delta]

L

[sinful herald]

I see, that is because \phi(1) should give you back the underlying operator A whose functional calculus we want to generate?

[quick pagoda]

Basically

By 1 I meant the identity function here, if you’re dealing with functions, the constant 1 function would give you the identity operator*

Since Phi(e)^-1 Phi(e)

[sinful herald]

yes, got it. Thanks again for help.

[void flame]

Could I get some help with this problem? I have no idea where to start, not sure what scaling solution i should use. Is there a specific form I should use?

[hearty wharf]

Are pdes wayyy harder than ODE

[buoyant pike]

Yes

[buoyant pike]

https://arxiv.org/abs/2503.01800

Big

[minor mulch]

macroscopic

[void flame]

I dont know what Hilbert's six problem is but sounds badass

[buoyant pike]

Rigorously justify various branches of physics mathematically

[queen elm]

remarkable

[junior bloom]

this is a very cool result but i need to think carefully about whether its the justification physicists are looking for. as ive mentioned, statistical physicists are suspicious of the hydrodynamic limit.

[verbal nebula]

Are there good reasons to be skepticial of the hydrodynamic limit?

Well, I'm sure there are, but do you have any insight on that?

[junior bloom]

viscosity changes as you rescale noise down. insight is hard, this is a very difficult physics problem.

among other things i think the first thing in this paper that a physicist would be skeptical of is the hard shell interaction, though i dont know if this is actually a big issue.

this physics paper was huge in theoretical phys

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.16.732

[verbal nebula]

Interesting. Seems above my paygrade for now

[junior bloom]

its a renormalization group problem if you know what that is

but this is generally why physicists, wrongly imo, dont interact with the whoole large deviations lit on hydrodynamic limits

i think physicists are phenomenologically correct but wrong not to interact because theres still some value

[verbal nebula]

Nope! I'm doing my project on inviscid limits and boundary layers. I'm learning fluids on the fly as I work on the research problems, so most of my knowledge is ad hoc at best right now

[junior bloom]

well heres the quick version. if you derive navier stokes from the boltzmann system in the standard physics way, you can imagine that the all the objects in navier stokes really come from coarse graining the microscopic description until you reach a continuum one.

the issue with the hydrodynamic scaling limit is that it's rescaling the dynamics in a vanishing noise regime, but when you coarse grain the objects from the microscopic dynamics to said scale, the "effective" viscosity keeps changing.

so the correct regime cant just have vanishing noise, the viscosity has to be limiting to something too in a coupled way with noise

what does it limit to? no clue

and maybe the solution is that mathematicians are correct in the end etc etc

but certainly the fixed viscosity argument does not make this make sense

[plucky pond]

can someone help me understand: if i have a PDE and want to express it as a differential operator P, how can I determine the semiclassical principle symbol? I understand that in the semiclassical case, the conjugate momentum variables \xi -> hD_x, where D_x = \frac{1}{i}\partial_x, but I am not sure how to treat the lower order terms.

the example I am referring to is the following operator P:
-h^{2} \partial_x^{2} - h^{2} \frac{A'}{A} \partial_x + (\frac{h^2k^2\pi^2}{A^2\theta_0^2}-1)
and we have Pu = 0.
Claim: the principle symbol is \xi^{2} + \frac{h^2k^2\pi^2}{A^2\theta_0^2}-1
But I am not sure why the lower order \partial_x term goes away, but the constant term doesn't

Thanks in advance

[quaint herald]

Unless I am misreading what you have written, the semiclassical principal symbol should instead be |\xi|^2-1. The reason the constant shows up is that any (hD)^\alpha, alpha a multi index, has semiclassical order 0. Terms are lower order in the semiclassical sense if the power of h exceeds the differential order. Eg h^2D has semiclassical order -1.

(But a scalar function, unadorned with semiclassical parameter is order 0).

[plucky pond]

okay thank you, so in this case, h^{2} \frac{A'}{A} \partial_x is semiclassical order h^-1, and \frac{h^2k^2\pi^2}{A^2\theta_0^2} is semiclassical order h^-2 ? (also, is there a notation for semiclasscal order, similar to bigO ?)

[plucky pond]

so if we had, say, P = -h^{2}\partial_x^{2} - f(x), then p(x,\xi) = \xi^{2} - f(x) ?

but P = -h^{2}\partial_x^{2} - hf(x), then p(x,\xi) = \xi^{2} ?

[quaint herald]

Remember h is small, so O(h), O(h^2) is more accurate, but there are various notations for the class of semiclassical diff/pseudodiff operators of a certain order.

[plucky pond]

so are you saying \frac{h^2k^2\pi^2}{A^2\theta_0^2} is more like order O(h^{4}) ?

sorry i just want to make sure im following

[quaint herald]

No looks like it's O(h^2) to me, i.e. of semiclassical order -2. There are no derivatives, but two powers of h.

[plucky pond]

oh alright, thanks

[quaint herald]

No problem :).

[plucky pond]

@quaint herald im just confused by what my professor wrote, as he wrote the principle symbol of
P = -h^{2} \partial_x^{2} - h^{2} \frac{A'}{A} \partial_x + (\frac{h^2k^2\pi^2}{A^2\theta_0^2}-1)
to be: \xi^{2} + \frac{h^2k^2\pi^2}{A^2\theta_0^2} - 1
it makes me think that A^{2}(x) (which in this case is a function that is included in the laplace-beltrami operator based on the metric on the specific manifold in which the problem is being analyzed) "cancels out" with the h^2 in the numerator by also being O(h^2)

and I feel I should mention that we are analyzing the PDE in a small strip of
area ~ O(h^2/3)

[quaint herald]

Without more context, I can't really say more. The domain being small shouldn't matter, but if A is like h^2 (which you gave no indication of earlier), indeed this term should be included.

If there is h-dependence of A, you should also check how the size of A' compares to that of A though, as for highly oscillatory A your first order (in differential sense) term could contribute to your semiclassical principal symbol.

[plucky pond]

ah okay, thanks again!

[zenith aurora]

Hi, I'm doing some research on Mathematical Models for Infectious Diseases and have so far a model for the time since infection(which instead of the usual 1 variable, it has 2 variables (a,t). where a is the time since infection and t is the 'normal/current' time
It's a System of 2 differential equations
One is a PDE and the other is an ODE. How would you go about doing some analysis for the model. I currently have a solution for the PDE but I'm not sure what else to include. If you want I can provide the exact equations .
(Also is this the best channel for this question ? )

[buoyant pike]

#modeling

[untold delta]

<Andrew>

[primal osprey]

Hey

[calm falcon]

Can someone help me with this problem? I imagine the solution involves first showing the inequality for a sequence of smooth compactly supported functions converging to u using an integration by parts argument but I'm unsure of the details

[rotund jetty]

can you integrate the hint by part somehow, like move an $x_i$ over from $u_{x_i}$ to the other stuff? I know it'll get you some second derivatives, but not sure if it will work. Also, I'd do the 1-D case first.

[untold delta]

L

[calm falcon]

i think the problem is the ^2, but i can move it outside the integral with jensen’s inequality

the resulting expression after IBP is p nasty tho

[calm falcon]

actually i think this is the right way to go. After moving the ^2 out and an application of Cauchy schwarz, you get an expression involving a product of powers of u, |Du| and |D^2u| unless im wrong, which suggests invoking Holder's ineq and dividing off the integral of |Du|

but the powers are a mess

[prisma marsh]

integrate by parts, note that the terms can all be controlled by $CuD^2u\left|Du\right|^{p-2}$, then apply generalized holder to that term with 1= 1/p + 1/p + (p-2)/p

[untold delta]

razor

[junior bloom]

can you guys think of any elegant expressions that can take a function, and return 1 if its first derivative exists and is nontrivial, and 0 otherwise.

this is a meaningless question since I can just define an operator that does this, but im after that maximum elegance

this essentially picks out the integer increments of Sobolev space index or Besov space regularity

that's why i feel like some nice expression exists for it

[astral vine]

If one has positive regularity index then W^{k,1} norms on balls are either finite or + infty. Since Wk'p embeds in Wk,1 Loc by construction.

You can replace Wk,1 norms by B^s_{1,+infty} to deal fractional index of regularity

But the case of Besov spaces requires to be on Rn at the first glance. On domains you need to use an equivalent norm that relies on finite differences instead.

[junior bloom]

Rn is fine, I'm doing physics, universe in a box etc etc

I can probably even make it periodic and take size to infinity

That's sharp. I'll look up those norms and use an appropriate exponential mapping.

Thanks!

I never quite internalized the importance on integrating on balls from harmonic analysis

[odd mauve]

Don't forget integrating on annuli, either!

[steady flicker]

so im trying to driving legendre polynomial's rodrigues's formula
notes says after eqution (24) i need to calculate (N+1)'s derivatives. But how to do this n+1 th derivative? I tried. but during the process I can't seems to find a pattern of some sort
how did the notes just got the line below?
Im struck on legendre polynomial for a day now. I stuck on this then I stuck on everything follows. Somebody help me please...
is this suppose to be take some order of derivatives and then find a pattern? than you do induction on n+1 th. but I can't find it

Look I know this is not suppose to be this channel. But I asked in normal pde channel and no answer. Im a bit desperate as I don't want to stuck and time is ticking. If anyone can guide me through spherical harmonics I would really appreciate it

[rotund jetty]

Can someone confirm: In this image, the $d\xi$ is the Lebesgue measure induced by the inner product on $T_x^*(M)$ given in coordinates as $(u, v) = v^T G^{-1} u$, where $G$ is the matrix of the metric tensor? So in case of the Laplacian, the integral appearing becomes the volume of the unit ball in $\mathbb{R}^n$ since $p(\xi) = \xi^T G^{-1}\xi?:

[untold delta]

L
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[quaint herald]

Which metric tensor? The setting is just a paracompact manifold equipped with a choice of density.

Great paper btw ❤️

Anyway, it's something like: you equip the cotangent bundle fibers with the densities such that when you tensor with the fixed density on the base space $\Omega$ you get the (canonical) symplectic volume on the cotangent bundle.

In local coordinates $(x,\xi)$ this density is $h(x)^{-1},d\xi$, where $h$ is chosen smoothly so that $\mu=h(x), dx$, where $\mu$ is the prescribed density on $\Omega$.

[untold delta]

grobmez

[rotund jetty]

I want to specialize the result to the setting of a compact Riemannian manifold and $P = -\Delta$. It seems that $h(x) = \sqrt{g(x)}$ and the integral $\int_{B_x},d\xi$ is always equal to the volume of $B_1 \subset \mathbb{R}^n$.

[untold delta]

L

[quaint herald]

Yep :).

[sweet osprey]

https://math.stackexchange.com/questions/5048655/are-polar-sets-lower-dimensional
i was wondering if someone here could maybe answer this?

[lament jungle]

Not sure which channel this best fits, but I have the following integral equation:
$$
c(x) = \begin{cases}
0, & x\leq 0\
\int_{t=-0.4}^{t=0.6} c(x+t) dt, & 0<x<1\
1, & 1\leq x
\end{cases}
$$
which I can transform by differentiation into this DE:
$$
c'(x) = c(x+0.6) - c(x-0.4).
$$

[untold delta]

ConfusedReptile

[lament jungle]

Problem is, I have no idea how to solve this. This is similar to a delay differential equation, but the method suggested there assumes the derivative doesn't depend on the "future" state, which mine does. This is a real problem, not from a textbook, so I have no more information.

I'm curious if there's a way to obtain an analytical solution. Any ideas?

[buoyant pike]

Hmmmm

Well

c'(0.4)=1

c'(0)=c(0.6)

In the range [0,0.4], c'(x)=c(x+0.6)

And in the range [0.4, 1], c'(x)=1-c(x-0.4)

[lament jungle]

And in the range [0, 0.6], c'(x)=1-c(x-0.4)
That's for the range [0.4, 1]

[buoyant pike]

Oh yeah

Typo

[buoyant pike]

So c''(x)=c'(x+0.6)-c'(x-0.4) so c''(0.4)=c'(1)-c'(0)=1-2c(0.6)

And c''(1)=-c'(0.6)=-(1-c(0.2))

And c''(0)=c'(0.6)=1-c(0.2)

Hmmmm

c'''(x)=c''(x+0.6)-c''(x-0.4) so c'''(0.4)=-2+2c(0.2)

This isn't going anywhere

[lament jungle]

I got a bit of an insight, I think. You can split the systen into 5 equal intervals of size 0.2 each. If you then call the functions in each region A,B,C,D,E, then the relationship between the regions reduce to a linear system
$$
A'(x) = D(x),\
B'(x) = E(x),\
C'(x) = 1-A(x),\
D'(x) = 1-B(x),\
E'(x) = 1-C(x)
$$
with boundary conditions between regions:
$$
A(0.2) = B(0), B(0.2) = C(0), C(0.2) = D(0), D(0.2) = E(0)
$$
and the old boundary conditions:
$$
A(0) = 0, E(0.2) = 1
$$

[untold delta]

ConfusedReptile

[lament jungle]

and this in turn can be rewritten as a fifth-order linear DE for, say, A. It won't be pretty though, I think, and I don't immediately see the solution from here

[buoyant pike]

Well, then A''(x)=D'(x)=1-B(x) and A'''(x)=D''(x)=-B'(x)=-E(x) and A^4(x)=D'''(x)=-B''(x)=-E'(x)=1-C(x)

And A^5(x)=D^4(x)=-B'''(x)=-E''(x)=-C'(x)=-1+A(x)

A(0)=0, A'(0)=D(0), A''(0)=1-B(0), etc...

[lament jungle]

there's another change of sign when taking E'(x), so A''''(x) = C(x)-1 and A'''''(x)= 1-A(x), I think

[buoyant pike]

Well a sign error isn't unlikely

Anyways

You can repeat this for B, C, D, and E

With 5 initial conditions for each

And solve each ODE

And then match the boundary conditions

[lament jungle]

Hmm, that's true. So it's theoretically solvable, I guess. I wonder if there's a way to figure out the answer without doing that, though, it really seems like the solution should be simple...

[buoyant pike]

Should it

You have a nonlocal ode

Anyways

For another solution technique

Consider a basis for L^2([0,1], f(0)=0, f(1)=1)

Plug each element in

See what happens

[lament jungle]

calculated the numerical solution and it seems to not even be continuous at the boundaries, so I probably overestimated how simple it'd be, indeed
EDIT: in hindsight it's obvious from the integral equation that it wouldn't be continuous at the boundaries

[buoyant pike]

Why is it bumpy

[lament jungle]

It's approximate, the bumps are just random noise.

[buoyant pike]

https://en.wikipedia.org/wiki/Delay_differential_equation#Another_example

Did you see this example

[lament jungle]

(the numerical solution is by simulating the random process that c(x) is the expectation for, which is "start from x and repeatedly add a uniform random number from -0.4 to 0.6, terminating if you reach >=1 or <0")

[lament jungle]

Ah, this is interesting. And not very reassuring, if simple-looking DDEs can have nonanalytic functions as solutions.

[buoyant pike]

You can try to do a change of time variable to get your thing into this form

You should also be able to solve the ode numerically directly

[lament jungle]

hmm, this form can't be obtained, I think, because left side needs to be a function of t while right side needs to be a function of 2t. so no linear transform on the time variable would do.

[buoyant pike]

This is what my numerical solution looks like

[lament jungle]

That doesn't look right (my plot should probably be correct). I also had weird results trying to get a numerical solution, it usually doesn't find a zero-loss one

As a simplified case, if one replaces -0.4 and 0.6 with -0.5 and 0.5, the problem becomes symmetrical, there's only two regions and it's analytically solvable. The solution can be written as
$$
A(x)= \frac{1}{2} \left(\sin (x)+\cos (x)+\frac{2 \cos (x)}{\tan \left(\frac{1}{4}\right)-1}+2\right)
$$
$$
B(x) = \frac{1}{2} \left(\cos (x)+\frac{\sin (x) \left(\sin \left(\frac{1}{4}\right)+\cos \left(\frac{1}{4}\right)\right)}{\cos \left(\frac{1}{4}\right)-\sin \left(\frac{1}{4}\right)}\right)
$$

[untold delta]

ConfusedReptile

[lament jungle]

with $A(0) = \frac{3}{2}+\frac{1}{\tan \left(\frac{1}{4}\right)-1} \approx 0.157102$, $A(1/2) = B(0) = 1/2$.

[untold delta]

ConfusedReptile

[lament jungle]

got an analytical solution in Mathematica, but the coefficients are for now too complicated to write down. The plot matches the numerical simulation, though, so it's likely correct. In particular, c(0) ≈ 0.397145 and c(1) ≈ 0.973242.

[buoyant pike]

Are you not enforcing c(0)=0 and c(1)=1?

[plucky pond]

can someone help me understand why, in practice, we may want to set the principle symbol of a differential operator equal to zero? let’s say i have an operator P = (-h^2 \Delta - 1), and some u such that Pu = 0. Should we set the principle symbol equal to zero due to the homogeneity of the PDE or is that mere coincidence?

[buoyant pike]

Well you get the symbol by taking the fourier transform of your equation right

[plucky pond]

@buoyant pike perhaps, see i always just pick out the highest order terms and replace the D^\alpha with \xi^\alpha

im not sure how to “formally” do it

[buoyant pike]

[plucky pond]

like if $$P = \sum_{\alpha \leq m } a_\alpha(x)D^{\alpha}$$

then the symbol
$$p_{0} = \sum_{\alpha = m} a_{\alpha}(x) \xi^{\alpha}$$

[untold delta]

shook

shook

[quaint herald]

The Fourier transform conjugates the action of P to the action of multiplication by p. One way of solving PDE is to essentially do the same thing but "dividing" by p instead. Of course usually we can't literally do this because of vanishing of p, but this should motivate why the zero set of p is important.

It is called the characteristic variety of the operator, and it's complement is called the elliptic set. For the Laplacian for instance the symbol is just |xi|^2, which is nonvanishing away from 0, which is precisely what makes it an elliptic operator.

[plucky pond]

so essentially the principle arrises because differentiating in space is like multiplication in frequency

[quaint herald]

Yep

[plucky pond]

is there a good resource you recommend that builds up these tools from scratch essentially

i think i have a lot of gaps in my knowledge here

[buoyant pike]

The wikipedia article for the fourier transform

[quaint herald]

Hard to tell your background and exactly what you are trying to learn. Something like Grafakos has a good systematic development of the Fourier transform, but less about the specific PDE applications.

Many PDE books should include use of the FT and concepts like the char set and ellipticity, e.g. Taylor.

[plucky pond]

thank you @quaint herald

[quaint herald]

No problem

[astral vine]

For people that are well aware about deep details of elliptic regularity. If I have an elliptic operator of order 2 on C^{1,a} domain (bdd) of R^n, 0<a<1. If I am able to prove that the solutions of the corresponding PDE on L^p does have W^{1+a+1/p+\eps,p} regularity, is it really impressive ?

Solutions are more regular than the boundary

lol

[pure fulcrum]

mb gang i got it

[buoyant pike]

[pure fulcrum]

lolz

im new to pde's world and when i came across the word separable it clicked my brain and i figured my things out

[winged torrent]

Would this channel be the proper one for asking about Z transforms?

[buoyant pike]

Perhaps

[upbeat spade]

we know that the robin bvp on the laplacian (with boundary condition del_n u = beta u) with an appropriate sign on the boundary coefficient beta is coercive.

on a bounded Lipschitz domain Omega

now, if we change the boundary condition to del_n u = i beta u then it satisfies a garding inequality. is there, for example, a range of values of beta for which it isnt coercive, or do i just have to find specific counterexamples? or maybe it is coercive and i just havent found the argument

actually, by typing out a terribly worded question i just realized how to solve the problem i was having

[cedar storm]

how do you apply spectral methods to odes?

Sorry if this is a tad off topic

[buoyant pike]

#numerical-analysis

Well

I guess here is fine

The idea is to consider a basis for some function space and see how the differential equation acts on the basis

If the differential equation is linear, then it will nicely split and the behavior on the basis will be all the information you need to construct solutions

[grave matrix]

When I think of spectral methods I think about assuming your PDEs functions can be written as a fourier series then taking DFT of the equation to find coefficients, and then inverse DFT to return to original domain

And the details are in how you interpret derivatives or do collocation

I never considered non Fourier basis functions

Is it still called spectral methods if you have non Fourier basis or dont take a DFT?

[buoyant pike]

Yes I would say so

[errant depot]

Given a function $f$ in $W^{1,1}_0$ is it always possible to consider a sequence in $W{1,\infty}_0$ that weakly converges to $f$ in $W^{1,1}$ ?

[untold delta]

Milo

[astral vine]

Yes.

Because smooth compactly supported functions are dense in W^1,1_0

[tired hollow]

Hello, can someone help me solve this ? h is the density function of V which is a stochastic process, and V overline is with V_t being equal to sigma square

[astral vine]

@quick pagoda the brain fart was insane actually

[quick pagoda]

Well, they are smooth in it too

[astral vine]

Yeah fair point.

[tall jolt]

Is PDE comparable to the subject of diophantine equations, in the sense that you can't say a lot from a general POV? Or is there more general structure than what diophantine equations have?

or less even

[verbal nebula]

https://math.jhu.edu/~js/Math631/klainerman.survey.pdf

This is a good essay

[noble gate]

if $G(x,y)$ is the green's function for the unit ball, i.e. satisfying:
$$
\begin{cases}
-\Delta_y G(x,y) = \delta(x - y) & B(0,1)\
G(x,y) = 0 & y \in \partial B(0,1)
\end{cases}
$$
how do i show that:
$$
\int_{B(0,1)} G(x,y) = \frac{1}{2n}(1 - |x|^2)
$$

[untold delta]

aNDY

[noble gate]

Is there a way to avoid having to do a direct calculation

[buoyant pike]

What have you tried

What does avoiding a direct calculation mean

[noble gate]

expanding G in terms of fundamental solution

[noble gate]

I'm trying the following approach

$$
-\Delta \frac1{2n} (1- |x|^2) = 1
$$
and
$$
\int_{B(0,1)} - \Delta G(x,y) dy= \int_{B(0,1)} \delta(x-y) = 1
$$

[untold delta]

aNDY

[noble gate]

so if i manage to show that
$$
-\Delta \int_{B(0,1)} G(x,y) = \int_{B(0,1)} -\Delta G(x,y)
$$
Using the fact that when $|y| = 1$, $G(x,y) = 0$, i have a poisson equation

[untold delta]

aNDY

[noble gate]

I mean morally speaking this is simple,
$$
v(x) := \int_{B(0,1)} G(x,y) dy
$$
morally satisfies:
$$
\begin{cases}
-\Delta v(x) = 1 & x \in B(0,1)\
v(x) = 0 & x\in \partial B(0,1)
\end{cases}
$$
so, by uniqueness,
$$
v(x) = \frac1{2n}(1 - |x|^2)
$$

[untold delta]

aNDY

[noble gate]

But, I don't know how to justifiy interchanging the laplace operator with the integral

I will move on from this question for the time being

[buoyant pike]

What theorems do you know about interchange of limit operations?

[verbal nebula]

The integral is against the variable y, but you're taking a laplacian in the x variable. Have you taken grad real?

[noble gate]

all the standard measure theory thereoms

[noble gate]

Yes

[verbal nebula]

I think Folland buries it in an exercise, but pick your favorite version of Leibiniz' Theorem/DCT that'll apply

[noble gate]

I'll bash my head against it for a bit more

[noble gate]

I just don’t think DCT applies because the limiting sequence blows up to a delta function

[verbal nebula]

The Theorem I'm thinking of is 2.27 on page 56 of Folland

Suppose that $f: [a,b] \times X \to \C$ and that $f(t, \cdot) : X \to \C$ is integrable for every $t \in [a,b]$. Let $$F(t) := \int\limits_{X} f(t,x) \mathrm{d}\mu(x). $$ Suppose that $\partial_t F$ exists and there is a $g \in L^1$ such that $|\partial_t f| \leq g(x)$ for all $(t,x)$. Then $F$ is differentiable with $$F'(t) = \int\limits_X \partial_t f(t,x) \mathrm{d}\mu(x).$$

[untold delta]

MoonBears-C-
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[verbal nebula]

The $\delta$ maybe singular, and such a $g$ may fail to exist in the way that you want, but you can always find an approximation to the delta, and take limits carefully

[untold delta]

MoonBears-C-

[verbal nebula]

Chapter 9 of Folland covers these details, but usually PDE people sweep such things under the rug. I know my professors, and my advisor does

There can also be good information in the appendix of Evans PDE book on these matters

[ocean ether]

what is happening from 5 to 6

[buoyant pike]

Chain rule

[ocean ether]

oh gg

[untold delta]

▇▇▇ ▇▇▇▇▇

[void raft]

Hey! Just wanted to see if I found the Fourier series properly, I'm not sure about the change in domain when finding the coefficients

[lilac barn]

#odes-and-pdes

[ocean ether]

anyways how can i pull 7 from 3 and 6

it's easy to see

$\sinh\xi\sin\eta\frac{\partial U}{\partial \xi}+\cosh\xi\cos\eta\frac{\partial U}{\partial \eta}=\cosh\xi\cos\eta\frac{\partial V}{\partial \xi}-\sinh\xi\sin\eta\frac{\partial V}{\partial \eta}$

from direct substitution but how can i pull $V$ from here

[untold delta]

▇▇▇ ▇▇▇▇▇

[tired hollow]

Ive heard monge ampere was used in calabi conjecture to prove existence of Riemann forms

Wikipedia

I kinda get the problem of prescribed curvatures

Is there any good background anyone know of

Cuz chatgpt wasnt very helpful

Mainly looking for description of how monge ampere and complex monge ampere are both used to solve for problem of prescribed curvature + how it was used in calabi conjecture

[buoyant pike]

Monge ampere show up a lot in diff geo

[buoyant pike]

For the problem of finding a surface of prescribed curvature K on a domain D in R^n, you solve monge ampere for z=u(x) on D

det D^2u-K(x)(1+|Du|^2)^(n+2)/2=0

The wikipedia article on the calabi conjecture explains where the complex monge ampere comes in

[tired hollow]

I guess more of wanting an explanation of how the equation actually corresponds to solving prescribed curvature problem.

I guess wikipedia does give references

Ty for pointing out second part tho

[wind mortar]

Did you type this out in latex?

[frank prism]

you can find a sketch of geometry -> monge-ampere on page 248 of Lee's Complex Manifolds book, and a full proof (all the analysis and such) as well as a discussion on how this relates to the curvature and holonomy of your manifolds in chapter 6 of Joyce, Riemannian Holonomy Groups and Calibrated Geometry (or his older book Compact Manifolds with Special Holonomy)

The very quick answer as to why it shows up, is you want a Kaehler metric in the same Kaehler class, which is really a cohomological condition on the Ricci and Kaehler forms, for which you can use the ddbar lemma to turn directly (in local coordinates) into the Monge-Ampere equation, where the ddbar lemma (a consequence of Hodge theory and thus the whole machinery of elliptic PDE) is what turns the conjecture, something involving curvature, so the metric and its second derivatives, i.e. n^2 equations, into one scalar equation

if you're not used to this neck of the woods with geometric PDE, the fact hodge theory has something important to say here (and in general, any time you have something involving curvature and/or cohomology on a compact manifold) is really the moral of the story (to me)

[tired hollow]

Much appreciated response ill definitely check out the section in lees book

[void raft]

Yes lol

[gleaming jungle]

Can someone help me with these questions?

[buoyant pike]

Have you tried any of them?

[gleaming jungle]

Yeah but I'm not too confident

I'm mostly interested in A

[buoyant pike]

Ok

[gleaming jungle]

If u know the answer

[buoyant pike]

What did you do

[gleaming jungle]

I substituted after decomposing, averaged in time and then rewrote and rearranged. I'm mostly looking for verification though

If you know the answer let me know then I can compare it

[buoyant pike]

If you post your solution I would be happy to verify it

[gleaming jungle]

do you also know the answer?

well i will show you tomorrow

cause im taking a break before sleeping

[sinful herald]

Hi, I am reading Chapter 6 on Schauder Estimates from following book https://www.worldscientific.com/worldscibooks/10.1142/6238#t=aboutBook. And I am stuck at above proposition. In particular I dont see how they obtained sequence g_m which converges in alpha-holder norm. Can someone please help me with this. Thank you.

[lilac barn]

Try mollification

[sinful herald]

I am not sure whether just mollification work or not, because I remember reading somewhere that C infinity is not dense in Holder space...

so I was thinking that some how the regularity of g is increasing but I dont know...

[lilac barn]

Start with approximation results on Wkp for bounded U. Refer Evans for that. Now use the embedding of Wkp into Ca via Morreys inequality.

[sinful herald]

but for that I need to start with Wkp function no?

[lilac barn]

Okay I was thinking you already have the mollification for Wkp. Otherwise, try proving that mollification indeed does give you an approximation.

You do need to make use of the regularity 2 exponent as it fails for 0

[sinful herald]

what regularity 2 exponent?

[lilac barn]

Oh g is just Ca?

[sinful herald]

yes...

at least that is what I think

this is the previous page. f is C alpha only

so I was thinking that g is also c alpha only but I am not sure if I am missing something

[lilac barn]

It should be fine if you pick b<a and require the convergence only in C0,b norm but yeah I think otherwise some information is missing

[sinful herald]

Ohh ok, so there is some typo ?

[tall jolt]

Is Terence Tao's "Nonlinear dispersive equations" apt for a first course in PDE?

[lilac barn]

Not really. The first should be Evans

[verbal nebula]

Folland has a good PDE book

[tall jolt]

would it be apt for a second course? I'm just trying to get a sense of the landscape of PDE books if that makes sense lol

[buoyant pike]

I think that it's relatively specialized

[tall jolt]

are Evans and Taylor roughly the same?

[astral vine]

absolutely not

In the end, one should read all the books: Evans, Brezis, Taylor, Tao, Folland, etc.

Because each gives an extremely different perspective and various an complementary ways to deal with PDEs

There are a lot of different books, but I would reocmmend Evans, Folland and Brezis first

Then Tao

Then Taylor. But one can start Taylor and deal with Tao during the read of the three books

[verbal nebula]

I've mainly read Evans, Han and Lin's elliptic PDEs, and Majda & Bertozzi for vorticity and incompressible fluid flow

I definitely feel that my PDE background isn't as broad as it could be, but I'm focused on getting through Majda & Bertozzi for now as that's more relevant

[astral vine]

Of course if one wants to be more specialized quicker in some subfield of PDEs, one may replace the books by some others. The one above, except Tao maybe, are quite general, somehow.

[tall jolt]

What interesting things can you say about Maxwell's equations from a purely mathematical POV? Either relativistic or non-relativistic, or on manifolds

[buoyant pike]

What is interesting to you

[tall jolt]

Math. Yeah so this question is not very specific. I don't know much about the "culture" of PDE. Given a differential equation, what kind of questions are natural to ask?

Maybe the Maxwell's equations have some regularity? When is uniqueness to be expected? How does the geometry of the underlying manifold affect the existence of solutions? Can you give some bound on the norms of the solutions?

[buoyant pike]

Well you can explicitly find solutions

Jefimenko's equations

[tall jolt]

ah, this is funny too

also, can solutions blow up or something? obviously not in physical situations, ig

[shell jackal]

kind of depends on what interesting means to you i guess but maxwells equations with no sources are exactly the yang mills equations with a U(1) gauge theory

if your base is compact kahler this is basically the same thing as asking that the connection is the chern connection of a holomorphc structure on whatever hermitian line bundle your equations live on

so in some sense understanding solutions to maxwells equations with no sources is the same as understanding holomorphic structures on line bundles on a kahler surface

[frank prism]

wait this is really cool, i somehow missed all of this in my geometry and PDEs education

[frank tide]

what book or level of education do you see this in? also what are the prereqs, diff geo and pde analysis?

[shell jackal]

I guess probably knowing what a holomorphic structure on a line bundle is

The statement about maxwells equations is very trivial in content it just follows basically directly from the reformulation in terms of the electromagnetic tensor (this is something any advanced E/M course will teach you)

well maybe its not that obvious that the tensor is the curvature of a connection...

Ok if you are working on R^4 then if you let A be the electromagnetic potential and express it as a vector 1-form then the definition of F from A just literally is dA

If you think about the transformation laws for the 4-potential under a change of coordinates you will see that it just literally is the transformation law for a connection so this is the right generalization to topologically non trivial spacetimes (i.e a 4-manifold lol)

when you interpret maxwells equations in terms of the potential and write everything out it just literally becomes dF = 0 and * d * F = J

so when J = 0 this is just the YM equations

If any part of this didnt make sense to you its probably just because you havent seen the relevant terminology. if you know what a connection/hodge star/the YM equations are nothing very deep is happening

[shell jackal]

yea i guess it is not really explained so much in the literature sometimes but this is basically how people came up with the YM equations lol. for U(1) they are maxwells equations, for SU(2) they model strong force interactions

The generalization of this is the donaldson-uhlenbeck-yau theorem which over a curve had a classical proof with algebraic geometry due to narasimhan-seshadri

[frank prism]

yeah I've seen all that (in like Wells' appendix), just never knew about the physics connections beyond knowing that they were lurking around a corner

[ocean ether]

what exactly is a "weak solution"

yeah sure we express it in terms of integrals (integral of this equals integral of that for all f in this space)

but what does it mean

[inland sinew]

I have question that is probably obvious but I feel uncertain about it for some reason. It's regarding a simple, linear first order pde of the form $a(x) z_{y} + b(y) z_{x} = 0$. More specifically, when the coefficients are discontinuous. I guess an example of this would be when a and b are equal to the sgn function, not defined at the origin. In that scenario, I was wondering if $z(x,y) = |x| - |y|$ is valid as a solution. In general I am not certain at all how to deal with even simple pde's like this when the coefficients are discontinuous, and whether or not I can even suggest solutions like the one I wrote.

[untold delta]

pewpew2385

[rotund jetty]

You can interpret the equation as an equality of distributions

[quick pagoda]

Well, what have you seen about this

[ocean ether]

ive just seen the definitions

[quick pagoda]

Well, have you seen the justification by spamming integration by parts?

[minor mulch]

i mean the definition literally is integration by parts

[ocean ether]

yes

but is there like

something we lose going from strong -> weak

[quick pagoda]

Yes

[quick pagoda]

Well yes but I mean the whole “a strong solution satisfies all this, so what if we look weaker by….”

[quick pagoda]

So, for one, you can have discontinuous weak solutions to some problems

[minor mulch]

https://math.stackexchange.com/questions/3314557/why-should-i-believe-in-weak-solutions-to-pdes

see especially the 3rd answer

and this one https://math.stackexchange.com/a/3315144/469384

[pulsar forge]

This is explained in the post linked as well, but essentially it means we are changing the way we looking at the PDE, and this motivation comes from the fact that we want a tool that matches what happens in reality.

Suppose you model some physical phenomenon using the PDE Lu(x)=f(x) with some boundary condition, well you can say mathematically that we are trying to find a function u that satisfies the above equation pointwise, and call it a day. But, as evidence shows, there may as well be physically accurate situations in such phenomenons, that dont satisfy your PDE because of some discontinuity or lack of regularity in the classical sense at certain points. You might say lets just require the PDE to be satisfied a.e , but that runs into uniqueness problems ofcourse, and as it turns out, weak solutions are the right way to handle this discontinuity problem, you are trying to formulate your PDE in a way that allows solutions to have discontinuity problems. Classical solutions are weak solutions, but you need to achieve some sort of regularity on your weak solution to say its a classical solution, by using sobolev embeddings for example.

Ofcourse this is really mainly interesting when discussing PDEs, since in the ODE case, you have that classical and weak solutions are equivalent under pretty weak assumptions.

[ocean ether]

ah ok

so the motivation is "irl doesnt need to follow strong solutions and weak solutions are sometimes fine for what were trying to describe"

[frank prism]

There's also the part where weak solutions can often be much easier to prove the existence of, and then as step 2 of your proof, show that such a weak solution must necessarily actually be a regular solution (i.e. prove it is smooth enough). so they're also very theoretically useful

[minor mulch]

PDEs frequently dont describe physics at small scales

i like to think of a PDE as a formal expression intended to describe some physical phenomenon

usually these are derived via some physical arguments involving approximations

so theres no reason a priori why a literal interpretation of the PDE would perfectly describe reality—it is a continuum approximation that was itself derived via other approximations

it’s more like “strong solutions are sometimes fine for what we’re trying to describe” not the other way around

even very well-behaved PDEs often don’t perfectly model the phenomena they purport to describe

for example the heat equation predicts that heat propagates faster than light

[lilac barn]

In the link quite a lot of answers are explaining the motivation behind studying weak solutions, but don't address the more elementary question of what a weak solution is?
To that end, first we recall a strong solution: it is simply a function which satisfies the PDE equality pointwisely: aka if you have n-derivatives in your equation, the function's n-th many derivatives should also exist pointwisely, and satisfy the required equality.

So the solution can be thought of as being defined intrinsically, i.e. I say what the function looks like and nothing else: pointwise definition or graph etc.

The inversion of this is defining the solution extrinsically, i.e. I don't say what the function looks like but how it behaves with others (A man is judged by the company he keeps). This leads to the integration-by-parts formula.
The reason this description can become weak is because I can limit the number of others with which I describe its behaviour.
That is, saying “f behaves with g like this” is weaker than “f behaves with g and h like this”.
And that is precisely what we end up doing: Instead of requiring the integration-by-parts to hold with the full dual space, we require it to hold only for test functions.

[void flame]

Could someone help me with this? This is for a takehome PDE exam I just submitted and I just couldnt figure it out 🥲 I think instead of writing the integral like this we should write the integral of t and the integrand being <G(x,t), phi(x,t)> since G is not expected to be a function just a distribution. But I couldn't proceed much further

I proved that it is a distribution but dunno how to show it satisfies the PDE

[exotic lava]

Try to get an integration by parts formula to move the t derivative from phi to G.
There will be a boundary term which is where the delta should come from

[tall jolt]

Can someone recommend a text on the Yang-Mills equations from a more mathematical perspective? Maybe something that's similar to Taylor's books

[unique stump]

over here, ℰ'(X) denotes the set of distributions with compact support on an open set X ⊆ ℝⁿ

bit of a vague question, but what's the significance of this theorem? like okay, i can write each such u in this form. so?

[pulsar forge]

I'm not sure if i can provide a satisfactory answer on the technical usages of this theorem, but on a more foundational aspect this is essentially a extension of the more general result that every distribution is locally the derivative of a continuous function. The importance of such a result is that if we want a theory that allows us to differentiate every continuous function as much as we want, no proper subset of the space of distributions can contain all continuous functions and be closed under differentiation, meaning we are being as economical as we can be with our theory.

This result tells you that for compactly supported distributions, you can ask for a bit more and transfer that local property to a global one by patching up your distributions by a certain partition.

[unique stump]

no proper subset of the space of distributions can contain all the continuous functions and be closed under differentiation

sorry, i don't understand this bit. can't we identify any continuous function with a distribution and use integration by parts to define differentiation?

[pulsar forge]

What im saying is that our definition of distributions is optimal for its purposes, because if we take any proper subset of the space of distributions, at some point the distributional derivatives of some continous functions would leave that subset.

[unique stump]

and by "transfer that local property to a global one", you mean the property of it being differentiable?

[buoyant pike]

Have you searched for any

[pulsar forge]

The property of being the derivative of a continous function. This is true for all distributions locally, but you need more than that to make it global.

well there is a similar global result in general, but its not a finite summation.

[unique stump]

i think i see what you're saying, but somehow the significance is still lost on me. do you tend to use this result when solving / proving existence of solutions to some pdes?

(doesnt help that this theorem hits a sort of dead end in the lectures i'm attending)

[tall jolt]

I was reading Hamilton's "Mathematical gauge theory"

I have yet to read chapter 8, but up to chapter 7 they mostly just introduced the framework for the equations and variations thereof and proved a few basic theorems (like corresponding variational principles, gauge invariance, etc.)

https://math.berkeley.edu/~gbeiner/papers/Yang_Mills.pdf this contains some results

[tall jolt]

where does Taylor define "variation"?

are they taking the "quantity" in question Q, depending on the metric g, and taking the derivative of Q(g+th) for all compactly supported covariant 2-tensor fields h, t>=0, at t=0?

[tall jolt]

well you probably need h to be symmetric too and nonnegative definite

but idk if that's what they mean

also, apparently, if the LHS is zero then G should be zero. But why is that the case?

[buoyant pike]

Off to #diff-geo-diff-top you go

[broken hamlet]

Donaldson-Kronheimer is also a reasonable source

Is there a good reference for non-linear flows? More precisely, the question I'm interested in is of the form
$$\dv{h}{t} = P(h)$$
Where $P$ is some non-linear (second order) differential operator. I can show that the linearisation of $P$ is elliptic. All the papers I read then claim at this point that there exists a short-time solution.

[untold delta]

shingtaklam1324

[broken hamlet]

I'm familiar with elliptic PDEs to the level of Evans or Gilbarg-Trudinger

[astral vine]

For most results of this type existence follows from a fixed point argument

So you are right wanting to deal with the linearization of P

In this case everythign boils down to semigroup theory

For Navier-Stokes say in L² type spaces, see this https://arxiv.org/pdf/math/0511213

Theorem 3.2

[shell jackal]

Donaldson kronheimer but you really need to be familiar with the full formalism of Fredholm theory for elliptic operators between bundles before hand

So maybe reading Lawson Michelson first would be good

[broken hamlet]

I see, for me the right hand side is from Yang-Mills theory, so it's only weakly elliptic (the diff op only has derivatives in some directions), but I'll have a look and see what I find

Actually since you're talking about this, @shell jackal do you know a good reference for short-time existence of Hermitian-Yang-Mills flow? I've tried reading Donaldson's ASD on Surfaces paper but haven't gotten much out of it

[tall jolt]

Thanks to both.

[shell jackal]

Sorry Im not sure

I can ask someone

[broken hamlet]

I see, no worries

thanks!

[sonic olive]

Any recommendations for sources to learn about the Weiner criterion for solvability of elliptic pde or even just laplace equation?

[quick pagoda]

I think Helm’s potential theory has it?

David-Semmes has stuff for uniformly rectifiable sets, which corresponds to SIO bounds, and hence the relevant L^p solvability stuff

[noble gate]

This might be too physical, but Evans’ talks about the breakdown of the Huygens’ principle in even dimensions, but then how do vibrations on drums not persist in an obnoxiously long way

[buoyant pike]

You'll have energy dissipation from various physical effects

[turbid scaffold]

For pdes is there any sort of theorem about the dimension of the solution space like there is in odes? For example what is the dimension of the space of solutions of the linear homogeneous pde (Dx)(Dx)+2(Dx)(Dy)+(Dy)(Dy). Or similarly, what is the dimension of the solution space of the laplacian?

[broken hamlet]

A priori there won't be such a bound, for example, the dimension of the kernel of a partial differential operator depends on the domain

e.g. on a closed manifold, any harmonic function is constant

which is absolutely not true in general

On the other hand, there are several related results

c.f. Fredholm alternative and spectral theory

[turbid scaffold]

Oh okay so for example on the plane there is an infinite dimensional space of harmonic functions so I know the laplacian doesn’t have a finite kernel. Thanks

[tall jolt]

on closed manifolds there's a formula for the index (kernel-cokernel) of elliptic operators, see Atiyah-Singer. In general, idk. Might it be possible to show that the dimension of the kernel of the operators you mention is always uncountable in R^n, n>=2? Obviously, it suffices to show this for n=2.

For the example you give, you can write it as (Dx+Dy)^2 and by a change of variables this amounts to (Dx)^2

[tall jolt]

I think it's possible to see this by considering power series. In fact, harmonic functions are analytic, although I don't think that's true in general. but idk.

[solid flint]

Hello,
Consider the following Cauchy problem $u_t=Au+f$, $u(0)=x$ where $x$ is in a suitable space (some interpolation space usually).
Is anyone aware of a reference for Maximal regularity in the case of Hyperbolic problems please? most of the results that I could find are about the Parabolic case, i.e. When $A$ is a generator of an Analytic semigroup. Any comment or input will be much appreciated. Thank you very much in advance.

[untold delta]

Mikahopff

[astral vine]

This is not possible in general

[solid flint]

Can you elaborate please? Does it work for some special Hyperbolic cases? any references please?

[astral vine]

The as I already gave you : Da Prato and Grisvard.

[solid flint]

Thanks, will check it

[buoyant pike]

References for well posedness for shallow water equations on a rotating sphere?

[tall jolt]

Can anyone explain the definition of a Friedrichs mollifier? What is varphi, and what is varphi(xi) supposed to mean?

[buoyant pike]

varphi is a function satisfying certain conditions, xi is in R^n

Do you know what a mollifier is

[tall jolt]

nope

so varphi is a Schwartz function on R^n

[buoyant pike]

https://en.wikipedia.org/wiki/Mollifier

read

[tall jolt]

varepsilon is a scalar I think. What is D_x? I'd guess that's partial derivative wrt x, but why can you feed it to varphi?

Ok. What did Taylor mean by C_0^infty? Compactly supported C^infty?

This came up when reading this (first screenshot). What is J_varepsilon ell_ij exactly? I know this definition (second screenshot) of convolution of a distribution and a function, but that again gives a distribution, not a function

[quick pagoda]

What’s this from

[quick pagoda]

Or decays to 0 at infinity

[quick pagoda]

I’d think it’s a derivative (or some other operator defined earlier), but you can apply varphi to \eps D_x by functional calculus stuff

[quick pagoda]

If the distribution arises from a function, that function is pretty unique

[tall jolt]

Taylor volume 3

[tall jolt]

Why should it arise from a function?

[quick pagoda]

Presumably proven somewhere in there, but it’s kinda convolution with a very smooth function

[rotund jetty]

yes

[rotund jetty]

I think since $D_x$ is multiplication by $2\pi i \xi$ in the Fourier domain, $\phi(\varepsilon D_x)$ is multiplication by $\phi(\varepsilon 2 \pi i \xi)$ in the Fourier domain.

[untold delta]

L

[rotund jetty]

(depending on the Fourier convention and exact definition of D_x)

[quick pagoda]

That probably works too

Should be the same

[tall jolt]

What is DF(w) and DF(0)? And why does it follow from the fact that DF(0) is invertible that (5.15) has a unique solution w with small norm, provided that H has small norm?

wait nvm I found it

But I'm not sure why the last paragraph is true. Could anyone explain it?

[rotund jetty]

It looks like a inverse function theorem argument, but I'm missing too much context to tell you for sure.

[lilac barn]

Convolving a distribution with a test function or tempered distribution with a Schwartz function gives you a smooth function

You can find the results in Follands real analysis book on section for distribution theory

[tall jolt]

oh okay yeah that's right, they are using the Banach version of the inverse function theorem. They do not explain it (I think), they just briefly mention that the inverse function theorem generalizes to this setting in chapter 1 lol

[tall jolt]

What are examples of domains where the Dirichlet boundary problem is not always solvable? Is there a characterization of the domains where the Dirichlet problem is always solvable?

[quick pagoda]

wiener had a criterion related to this didn't he?

[tall jolt]

Yes. Do you have a prefered reference?

[quick pagoda]

No but Helms is the first one that comes to mind

[tall jolt]

This is done in Taylor chapter 11 too

section 6

[quick pagoda]

Ok cool

[astral vine]

Take the unit disk remove a sector of angle theta

There is condition depending on theta for existence and by duality for uniqueness

[tall jolt]

what's this condition? And also, if there are solutions at all, aren't they always unique? (it reduces to boundary condition =0 then use maximum principle)

[astral vine]

Is for you the Dirichlet problem always considered on L², if so yes. Otherwise on Lq no.

[sonic olive]

Do you have any good sources to learn how to solve elliptic pde with L^p boundary data?

[astral vine]

I know references, but good ... Not anything sufficiently pedagogical to really learn by yourself as fresh graduate student

[waxen bobcat]

share them anyways

[sonic olive]

What would be the easiest to approach then? lol

[potent sphinx]

Need a good RIGOROUS THEORETICAL book on ordinary differential equations that does all the proofs
And followed by a theoretical PDE book
I prefer if it uses differential forms approach ie it frames differential equations as functional equations between differential forms interpreting all the dx dy rigorously as differential forms

[potent sphinx]

Personally I want something
MOST books don't even properly tell what a differential equations is rigorously compared to our known real analysis
Just the initial value problem that's all
I want something to interprete something like xdy-ydx=0
As a functional equations where the x ,y are functions
The dx dy are first order differential forms
And relations between them
We need to solve for x and y
With a given initial condition

[heady silo]

cant you just look at coordinates of the form and get an equivalent formulation?

[potent sphinx]

Huh
I dont get it

Is there no book that has what I want
Ie interpreting a differential equations with the dx dy dz as a functional equation of differential forms of various orders

[heady silo]

if $x_i$ are your coordinates then $df = \sum_i \frac{df}{dx_i} dx_i$, and two forms are equal iff their components are equal, the components of $df$ are $\frac{df}{dx_i}$

[untold delta]

wasd

[heady silo]

so from any equation of differential forms you can always get an equivalent formulation in terms of PDEs

by looking at the components

at least you can get local solutions this way, for global solutions when a global coordinate system doesnt exist on your manifold you need some other methods I guess

[heady silo]

for example x dy - y dx = 0 if x is your coordinate then would be x y' - y = 0 which has solution y = c*x, c constant

[undone sable]

the equation $x dy - y dx = 0$ is false as an equality of differential forms. It's not meant to be interpreted via differential forms. It's meant to be interpreted as $x \frac{dy}{dx} - y = 0$

[untold delta]

L

[undone sable]

differential forms are used in other ways in ODE/PDE

[chrome trout]

Differential forms are things from differential geometry

And I am not sure how do you say that two differential forms are the same, since they are (in this case and most of the time) defined locally

You could say that those functions $\frac{\partial f}{\partial x_i}$ are the same locally but what about outside of your chart?

[untold delta]

Mnp

[astral vine]

a differential form with distributional coefficients on an openset of Rn are just the data of properly indexed {2^n} R-valued distributions.

Since this is true on an openset locality is not an issue.

[chrome trout]

[astral vine]

Haha.

Do Carmo's approach forever in my heart.

[heady silo]

you can define differential forms in a coordinate free manner, like how tangent vectors can be defined without coordinates, forms are simply antisymmetric tensors. alternatively you can choose charts that cover your space and say that two forms defined on two different charts agrees on the overlap if they transform into one another under the transition map between the charts, then you can say what you have is a genuine form on your entire space

[heady silo]

it is a genuine equation of forms, if you say y is some function in the x coordinate on the space R then you can derive your equivalent formulation from this equation

[rotund jetty]

Assuming x and y are coordinate functions, then x dy - y dx is not the zero form because the coefficient functions x and y are not both identically 0

But if you pull back the form, say by a curve, then it is an ODE and you can get 0

[heady silo]

yeah definitely but if y(x) is a function on R then it is an equation equivalent to the ODE x * y' = y

if x is the coordinate

[rotund jetty]

yeah this can be framed as pulling back by curve gamma(x) = (x, y(x)). After pulling back, dy = y' dx so the equation becomes xy' - y = 0.

[chrome trout]

yes you can; they are sections

[heady silo]

frfr

[sinful herald]

Hi, Dose anyone know a reference for book which gives physical intuition for Helmholtz decomposition (and fluids in general), I can see it in R^3 this decompose vector field into a divergence free and curl free part, but I want to know what curl free part signify . I am sorry if this is not correct place to ask this reference.

[buoyant pike]

For fluids, when you do the helmholtz decomposition, the decompose the velocity into the div free and curl free components

The div free component corresponds to vorticity/the fluid stream function

And the curl free component corresponds to fluid divergence/the fluid velocity potential

[sinful herald]

I see, thanks. Do you know any reference where I can study more about these?

[buoyant pike]

Standard fluid dynamics books should cover this

[rain phoenix]

david tong has some notes on fluid dynamics i think are worthy of checking out

[verbal nebula]

Majda and Bertozzi is a good, yet imperfect book on Vorticity and Incompressible Flow is good but imperfect
There's also the book by Farbi on Analysis Tools for Fluids which is very good.
Galdi has a book on Navier-Stokes, which is also very good

I'm spending one year reading Majda & Berotzzi, then one year on Farbi's book, then one year on Galdi. (In conjunction to my research and attending seminars).
It's been interesting

[buoyant pike]

Well Majda/Bertozzi does not talk about fluid divergence

Because incompressible flow lol

[astral vine]

Galdi is too much imprecise and claim stuff without proofs saying just "should be easy/straightforward" while a lot of tiny (sometimes, non-)critical technical issues are involved. Some of them are not even known by most specialists.

[unborn gyro]

For fluids, I think Vicol and Bedrossian is excellent

[mortal slate]

Question

Suppose I have two functions which agree as distributions

Do they agree as functions?

[shy narwhal]

I think https://en.wikipedia.org/wiki/Fundamental_lemma_of_the_calculus_of_variations#Versions_for_discontinuous_functions is what you want

[mortal slate]

Thanks

[fierce forum]

Tong's notes would give some idea but it's not an expert material. The references in his notes are worthy reading though

[rain phoenix]

I mean

I think they're good for introductory purposes

If you've got a good set of notes I'm all ears!

[verbal nebula]

Eh it'll be fine. Have you taken a look at Farbi?

[lilac barn]

Vicol and Bahouri should be enough. Moreover, Bahouris book has the proof in Besov spaces which should be enough for all intent and purposes

[coarse sorrel]

did not understand step 6(last line) and step 7 full part

[lilac barn]

For step 6, they literally just plugged in the expression and gathered the terms for delta tf

[astral vine]

I don't know any Farbi, do you have a name ?

[verbal nebula]

Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations andRelated Models

Pierre Fabrie and Franck Boyer

My spelling is getting worse as I read less literature

[astral vine]

Ok this one I know.

[sinful herald]

Thanks for reply (and sorry for late reply), I am also Reading Galdi, but it dose not give me physical intuition most of the time, and unfortunately I am thinking that I am missing a lot because of that.

[sinful herald]

I agree its sometimes very hard to fill the details, and a lot of time I am not able to...

[sinful herald]

thanks a lot this seems useful. I will check that out

[sinful herald]

Thanks, I will see Bahouri book, Vicol's book I have known about but not read, I will check that out. Thank you

[verbal nebula]

Any thoughts on it?

[sacred cliff]

any recommended books? I haven't touched pde since undergrad long time ago

[shy narwhal]

depends on your analysis background but evans

[astral vine]

I would say this is mostly French Galdi.

The overlap is huge just it has a different presentation

Galdi is a little bit more versatile, while Boyer and Fabrie are a bit more detailed.

[turbid scaffold]

I’m confused by the nomenclature in quantum mechanics.

Are all solutions to a Schrödinger equation called wave functions?

[river path]

a solution with initial data that has L^2 norm 1

alternatively, a ray of solutions

[turbid scaffold]

Oh a wave function is a solution with certain conditions?

[river path]

a wave function must describe the state of a quantum system as a probability distribution, in particular its norm squared must have total integral 1

so yeah

[serene mica]

I was reading this paper https://arxiv.org/pdf/2402.07534 and the author produces solutions to the stationary navier stokes equations in BMO^{-1}(T^2). Personally I am not very familiar with this space, and google isnt giving much. Wondering if anyone knows what this space is? ||Bonus points if you can identify the point in the paper the author actually shows the solution is in this space 😂 kidding...kinda...||

In the same way the dual space of H^s is H^{-s}...maybe the dual space of BMO? Total guess though

[lilac barn]

Yes, as the naming suggests ( BMO ^{-1} ) is the dual space of (BMO) functions. More simply, you can think of it as distributional derivatives of (BMO) functions. You can find some mention in Tsai and Bahouri's book. It can be shown that
[ B^{-1}{\infty, \infty} \subset \dot{B}^{-1}{\infty,\infty} \subset BMO ^{-1}
]
and the author of the paper end up proving that ( u \in B^{-1}_{\infty, \infty} ) I believe.

[untold delta]

cocat

[serene mica]

Thank you kind sir 🫡

[astral vine]

BMO^-1 is not the dual of BMO functions, however your second description is correct.

It can be identified with div (BMO)ⁿ

And is the dual of the (homogeneous -on the torus this is not really important-) Hardy Sobolev space H^{1,1}.

[lilac barn]

So is the dual of BMO known?

[astral vine]

Not in my knowledge. But what you can do is compute the bidual of the Hardy space H¹ using "Riesz transform over L¹" characterization. This gives us by retraction/co-retraction argument that the dual of BMO is the vector space of finitely additive signed measures such that their Riesz transforms also are.

Which is super ugly and unusable in practice, I think.

[astral blade]

what are some good reference texts for learning de giorgi-nash-moser theory? i've only seen some short exerpts on it, but nothing particularly in depth. i assume there are more extensive texts for it that are not the original source papers (but please correct me if i'm wrong). thanks!

[astral vine]

An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (Publications of the Scuola Normale Superiore, 11, Band 11)

Very good even if it contains a certain number of typos

[cloud thistle]

Please don't post the same question in multiple channels simultaneously

[astral blade]

Thanks!

[quick pagoda]

What’s up with semiclassical symbols (and related things)?

[quaint herald]

They measure the "leading order" part of a semiclassical (pseudo)-differential operator both in the differential sense and in the sense of size as the parameter h->0 (when applied to functions/distributions oscillating at scale h).

[quick pagoda]

[noble gate]

i need some pde practice problems i have a final tmr on evans chapters 1-4, 6, 8

Primarily focused on 6 and 8

and also chapter 4

the exam is focused mostly on 6 and 8, but i'm worst on 3 and 4

[steady flicker]

this is from Evans, page 369, I got (a) and (b), how to show (c)?

[noble gate]

have u computed the expressoin for each

[steady flicker]

I don't get this take derivative in $\frac{x}{|x|}$ thing

[untold delta]

yehuihe

[steady flicker]

im not that good at this

previous problem just express in laplacian in spherical coordinate and Iget it. but what is $w_r$ expression?

[untold delta]

yehuihe

[noble gate]

it says it below, no?

$$
w_r := Dw \cdot \frac{x}{|x|}
$$

[untold delta]

aNDY

[steady flicker]

But how to differentiate this expression?

[lilac barn]

It is a straightforward calculus exercise. For (w:=e^{i \sigma \omega-x}), we have
[
r w_r-r i \sigma w=r e^{i \sigma \omega \cdot x} \frac{i \sigma \omega \cdot x}{|x|}-r i \sigma w = \frac{r}{ \lvert x \rvert} w ( i \sigma \omega \cdot x ) - r i \sigma w=r i \sigma w ( \frac{\omega \cdot x}{ \lvert x \rvert} - 1)\nrightarrow 0
]
as (x) could be (x \perp \omega). But for (w := \frac{e^{ i \sigma r}}{4 \pi r }),
[
r w_r - r i \sigma w=r \frac{i \sigma r-1}{r} w - r i \sigma w=-w = - \frac{e^{ i \sigma r}}{4 \pi r } \rightarrow 0.
]

[untold delta]

cocat

[steady flicker]

ok i got it. thank you.

[tropic rock]

can someone tell here the books the you used often use in research and reading and using as reference for pde beside evans book

[quaint herald]

Hormanders 4 volumes, Taylors 3 volumes, Taylors book on pseudo differential operators. Plenty more that are more narrow in focus.

[tropic rock]

thank you. If other can add more, please do add

[tropic rock]

I haven't started studying PDE, but may be can you add something on nonlinear parabolic pde type

as well if those doesn;t cover

[astral blade]

I found Gilbarg and Trudinger to be helpful

Kato discusses partial differential operators if that's something you're interested in as well

and operator theory broadly

[astral vine]

Before reaching non-linear parabolic PDEs one should be aware about linear theory.

Even then there are dozens of way to deal with parabolic PDEs (linear or not) depending of your goal problems

Some overlap, some are absolutely singular away from the others.

[tropic rock]

Then can you please list book reference on linear theory

Thank you

[astral vine]

Provide more info regarding what I said after.

[tropic rock]

unfortunately i am not familiar with those specific details, but based on one paper assigned for reading to me
"singularity formation, blow-up dynamics, and the asymptotic behavior of solutions in nonlinear PDEs" are potential topics

[astral vine]

Who are the authors ?

[mint canyon]

Is it normal for the Polar equations of an Exterior differential system associated to some PDE to be underdetermined?

[astral vine]

If you count the number of equations and unknown you will see this does not match

[mint canyon]

Yeah, Im tryin to do 2 dimensional laplace equation, it seems to have 7 unknowns for the Polar equations of the second integral element

I was advised to treat the unknowns for the independent variable basis as constants, and all the others as functions

(7 unknowns and 3 equations*)

[astral vine]

number of scalar equations

[buoyant pike]

#odes-and-pdes

[proud panther]

Oh sorry

My bad I didn't see that...

[crisp mango]

Is pde really that big

Ode
Pde
Laplace transform
Fourier series
Fourier transform
Z transform

I just have this topics exam isn't it very easy to checkup the process and solve why I stuck in exam? I practice hell lot

[waxen bobcat]

#odes-and-pdes

[pulsar basalt]

is this where you live?

[quick pagoda]

Or foundations channel

[pulsar basalt]

thats hell

[broken hamlet]

Is something which locally looks like a differential operator a differential operator? More precisely, Let $X$ be a manifold, $E$ a vector bundle on $X$. Choosing connections on $X$ and on $E$, we can make sense of the $k$-th derivative $\nabla^k u$ of a section $u$. A differential operator of order $k$ is a map
$$P : \Gamma(E) \to \Gamma(E)$$
of the form $P = P(u, \nabla u, \dots, \nabla^k u)$.

On the other hand, say we had a map $Q : \Gamma(E) \to \Gamma(E)$ which under any trivialisation of $E$ gives a differential operator for maps $s_1, \dots, s_m : \mathbb R^n \to \mathbb R^n$, i.e.
$$Q_j = Q_j(s_i, \nabla s_i, \dots, \nabla^k s_i)$$
Must $Q$ be a differential operator, as in the above definition?

[untold delta]

shingtaklam1324

[broken hamlet]

Actually I think the answer is yes, since "being a differential operator of order k" is independent of the choices of connections, since the difference of two connections is just an endomorphism (ie no derivatives)

[weak veldt]

i been revising for this exam, and as my understanding for this class increases, i am finding it so much cooler

especially so compared to any other class i have taken

pdes are powerful

and studying them abstractedly is too sexy when it fits into my brain correctly

[digital ibex]

For those aware of the Tau Method, how many tau terms do we need for a 2nd order PDE?

(I really should be asking about an order N pde but i just want this as a base for now)

Should the number of tau terms equal the number of equations (when reduced to first order) or should it equal the number of boundary conditions?

Im seeing conflicting answers online

[flat crow]

I know that for navier stokes, weak solutions are weakly continuous in $L^2$. However, is it true that in 2D, weak solutions are in fact strongly continuous in $L^2$?

I believe it is, for similar reasons as to why the energy equality holds

[untold delta]

jamiecjx

[flat crow]

I'd just like to find a reference in a book to connfirm this

[astral vine]

For 2D Navier Stokes we have Global well posedness.

However, equality energy is not good enough to warranty such results as such, see 3D Navier Stokes

It works because of 2D and the subsequent Critical Sobolev embeddings.

[flat crow]

Yeah, makes sense

I also just found an explicit mention of continuity in H for 2d weak solutions in the book I was using as well

(Temam navier stokes and numerical analysis p294)

[lilac barn]

Another Reference: Vicol Theorem 3.4

[arctic whale]

This is a lemma used to prove the oscillation theorem for De Georgi - Nash result.

Isn't this just the usual poincare inequality? Why is there a positive measure assumption for it?

[quaint herald]

Standard forms of Poincare either require u to vanish on the boundary, or have mean zero, or something morally equivalent. This first assumption in this lemma plays a similar role, and you need something like this.

Otherwise just look at constant functions for a trivial counterexample.

[lilac barn]

To give a more precise meaning to the above comment, look up the proof of Evans for Poincare inequality. At the end, he claims that v (which is the subsequential limit of a normalized contradicting sequence) is constant. Here the extra assumption pops into imply v = 0 which contradicts v being unit-normed (being the limit of a unit-normed sequence).

[arctic whale]

Yeah that makes a lot of sense; I thought they were talking about u in H^1_0. Thanks a lot

[arctic whale]

I am more aware of the proof by Riesz potential but I will look Evans up.

[dull dagger]

@raven shore

[frank tide]

@raven shore

[onyx inlet]

Why are yall pinging this user

cat mog do you know these people?

[raven shore]

Friedrichs inequality

[onyx inlet]

what.

[raven shore]

Lax milgram.

[onyx inlet]

Okay, I am just going to remove all three of you from the advanced channels. Feel free to appeal in @limpid slate if you feel like actually explaining anything.

[raven shore]

[graceful basin]

Hi guys

Anybody???

[shy narwhal]

hello fellow PDE student

[buoyant pike]

[dire anchor]

I’m trying to look at algebraic classification of simple differential equations—such as ODEs or classic PDEs (wave, heat, Poisson)—from an algebraic viewpoint rather than the usual analytic one. Could you suggest some key phrases / topics I should search for? (So far I’ve come across algebraic analysis, D-modules, differential rings/differential Galois theory and symmetry/Lie-group methods.)
What matters more in practice? D-module machinery or Lie-symmetry / group-classification?

[buoyant pike]

In practice for a pde theorist, none of this matters

[undone sable]

there are the elliptic, parabolic, and hyperbolic classifications, which are sort of algebraic

[zinc mortar]

John Pardon has managed to find an $\infty$-categorical context for the moduli space of solutions of linear and non-linear PDE that gives the moduli space better formal properties, or so I'm told.

[untold delta]

NotKnot12

[zinc mortar]

At the end of the day I think the goal is to avoid unwanted singularities and general bad behavior of moduli spaces that appear in the classical setting when you are trying to get curve counts in symplectic geometry, but again not super familiar with the details

[frank prism]

requires a decent amount of differential geometry, and not very useful for simple DEs, but exterior differential systems (also called Cartan-Kaehler theory) is a very useful approach, essentially recasting solving your DE in terms of geometric integrability problems (Frobenius' theorem from smooth manifold theory is the simplest such example) and differential ideals within the exterior algebra

[dire anchor]

Yes ultimately I am looking for geometric and topological interpretations of DEs, for example I rarely see hyperbolic PDEs on Riemannian manifolds.

[buoyant pike]

Bruh people solve all sorts of PDEs on manifolds

Hyperbolic PDEs on various manifolds is literally all of geophysical fluid dynamics

[celest fiber]

Related to electrostatics but it's a math question.
When solving for boundary conditions on sphere and the condition isn't a Legendre Polynomial is there a way to calculate the general expression for the series coefficients?

I've had this problem where a sphere is placed on the surface with $V0 cos(3\theta )$ and we know that the solution is then the infinite sum
$$ \sum_{n=0}^{\infty} A_n r^{-(n+1)} P_n (\cos (\theta ))$$
(where $P_n (x)$ is the n-th Legendre Polynomial).

i then found the first few terms of the expression using the orthogonality of the $P_n (x)$ (since $\cos(3\theta ) = 4\cos ^3(\theta )-3\cos(\theta ))$, but I wanted to know if there's a way to compute all the coefficients like one does in a Fourier series, (like getting a closed expression with respect to n).

[untold delta]

Henry_quite_hungry

[buoyant pike]

https://en.wikipedia.org/wiki/Legendre_polynomials#In_trigonometry

Using the legendre <-> chebyshev poly relations is porbably the easiest thing to do

You can also do this with integrals but you need to be careful about normalizing the inner product on the surface of the sphere

[quaint herald]

Uh oh. Several of my research projects are in trouble if the wave equation on Riemannian manifolds is a lie.

[buoyant pike]

Uh oh

Better stick to the heat equation

[dire anchor]

Sorry I'm not familiar with geometric analysis nor PDE

Just saying that from what I've known so far. Well the Jacobi field equation has the Sturm–Liouville form of wave equation

[quaint herald]

@buoyant pike technically on topic for this channel if we consider maxwell's equations and the recording of the visible spectrum for artistic purposes, but I am currently fighting the urge to buy a supertele zoom lol. discovered an incredible wildlife conservation island/park near me and the bird opportunities were insane.

[buoyant pike]

See if you can figure out a way to record sound from long distances

(Euler equations)

[steady flicker]

guys, im doing a derivation, but condition I'm using is regarding asymptotic form, thus has O(x^{-1) term. if I take the derivative, I need to do something about derivative of O() term right? I'm omitting it at the moment but seem I need to specify in the paper

[undone sable]

yeah, you should probably cite asymptotics for the derivatives of the Bessel functions instead. I would use Mathematica to get the asymptotics, and then find a reference later

[vast sedge]

Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory https://arxiv.org/abs/2503.01800

"In this paper, we rigorously derive the fundamental PDEs of fluid mechanics, such as the compressible Euler and incompressible Navier-Stokes-Fourier equations, starting from the hard sphere particle systems undergoing elastic collisions. This resolves Hilbert's sixth problem, as it pertains to the program of deriving the fluid equations from Newton's laws by way of Boltzmann's kinetic theory. The proof relies on the derivation of Boltzmann's equation on 2D and 3D tori, which is an extension of our previous work"

[weak veldt]

🤯

[steady flicker]

But almost all books I checked they give you 2.22, 2.23 but never want for derivatives of the asymptotic form

[vapid wharf]

https://arxiv.org/abs/2504.06297

Important comments

Found it from a John baez bluesky thread

[undone sable]

there is a recurrence relation that expresses the derivative of J_n in terms of J_n, and from it you can prove that the derivative of J_n(x) has the following asymptotics:

[slate summit]

Does anyone know of an interesting problem is differential equations which require non-metrizable topological spaces?

[undone sable]

distribution theory

[slate summit]

could you be more specific?

[pulsar forge]

the space of distributions is not metrizable

[rotund jetty]

The space of distributions on an open set $\Omega$, denoted $\mathcal{D}'(\Omega)$ has a topology that is not metrizable.

[untold delta]

L

[waxen bobcat]

Doesn't D(Omega) usually denote the space of test functions?

[rotund jetty]

yeah I forgot the '

[quick pagoda]

I don’t think the weak topology in L^p is metrizable

(The unit ball, perhaps, but…)

[shell jackal]

the weak topology on an infinite dimensional normed space (over R or C) is never metrizable

[heady silo]

why

[quick pagoda]

Try it

I mean, play with it and try and see why it fails

[shell jackal]

Essentially you ||take a countable basis U_n at the origin induced by some set of functionals F_n, finite for each n, and show that their union spans the dual of X. But then you have a countable basis for the dual of X, so it's in fact finite dimensional||

||the union of the functionals span, I mean||

[wary elk]

This is a standard result in functional analysis and you can search for a proof either online or in textbooks. I vaguely remember that open sets in the weak topology are unbounded so if the topology is induced by a metric, you would get the contradiction that a weakly convergent set being both bounded and unbounded.

[quick pagoda]

Doesn’t have to be just one argument, but yeah it boils down to open sets have affine subspace or wtv

Me when the kernels of functionals

[heady silo]

Im happy I know what the weak topology is, feels like im part of the gang now 😎

[shell jackal]

it is good

[unborn cedar]

Im a starving child starving to death and this is the very last thing i have
ever read. Gootbye

[river path]

im going to steal and publish this

[tired hollow]

lol I have a lot of proof that it’s my work

[green schooner]

Can you remind me how the Leray projector works in simple terms?

[tired hollow]

It removes the pressure term so the equations are about velocity only

[rustic oak]

Why are we engaging with this

[tired hollow]

because it makes sense that if you can show all valid inital data is in Schwartz space and you can show Schwartz is closed under Navier Stokes evolution, and you can show the seminorms remain finite and bounded and that the solution is in Schwartz, you can show, by definition, the solution exists and is globally smooth

Why wouldn’t you engage with it? Anyone who calls themselves a mathematician should want to engage with this

Anyway.. thanks to anyone who does want to engage on this

[rustic oak]

who Elon musk claims is better at math than the average PhD
Only if better means being able to recite facts that can be googled

[unborn quiver]

Has someone with an actual math PhD confirmed this claim

[rustic oak]

Every other week someone "solves" Collatz or Riemann or some other famous pop-math problem

And then proceeds to demonstrates zero knowledge of actual math

[tired hollow]

I understand why you’d be skeptical.. but I think this is real. Solid. Good math

[unborn quiver]

Although this result would be big, a 3 page paper with 1 citation is insufficient for the detail necessary to prove such a big claim (i.e. you can't get away with hand waving here)

[tired hollow]

I agree that’s probably true. Unfortunately I don’t have any community involved with math. I have no humans to talk to about math. Nobody I know knows math.. so I talk to AI. So now I’m bringing it here.. hoping for genuine human feedback

I think it’s legit

[median forum]

The problem is in 5. You don't actually establish any of these bounds

[unborn quiver]

Yeah, the claims of "can be..." don't provide any explicit details in the matter

[quaint pumice]

we don't care what grok has to say it's a stochastic parrot and all current LLMs are sycophants

[shy narwhal]

🦜s are cute

[quaint pumice]

it's not usable to verify proofs in intro classes much less for a millennium problem

[rustic oak]

The easiest way to spot an AI generated proof is
[trivial step]
[trivial step]
[trivial step]
[trivial step]
[highly non-trivial step that's the hardest part of the proof but provides zero elaboration other than claiming the step is possible]
[trivial step]
[trivial step]

[tired hollow]

This is what gronwell inequality is for

[median forum]

The hard step is getting to gronwall

[tired hollow]

Ok so expand further on section 5?

[median forum]

i guess, currently the first step of the estimate involves taking a derivative inside a supremum which seems sketchy without justification to me, and the rest just doesn't exist

[buoyant pike]

Are you saying that https://www.pnas.org/doi/full/10.1073/pnas.2500940122 is wrong then?

[meager dune]

Where is existence mentioned

[tired hollow]

It follows from the closure of Schwartz space under Navier Stokes evolution

[lilac barn]

What is assumption (4) in the paper?

[tired hollow]

Equation (4) is the decay condition given in the problem statement. It ensures the initial conditions decay faster than any polynomial. Given that condition you can show the initial conditions live in Schwartz.

[tired hollow]

No, my understanding is that it actually complements this result. Their paper shows that blow-up can happen when viscosity is zero (Euler equations), while mine proves global smoothness assuming viscosity is positive (Navier–Stokes). Viscosity is what introduces enough dissipation to control the nonlinear term—it’s the reason we get a negative-definite contribution in the seminorm evolution, which allows us to bound all derivatives globally in time

[median forum]

Your paper just doesn’t establish this bound though. I’m trying to engage in good faith here but it’s hard when you just say things are untrue

[tired hollow]

It does. we take the time derivative of the seminorm, show its bounded above by a function of itself, and apply gronwell inequality to establish the seminorm is globally bounded

Plus consider the key insight. Initial conditions necessarily live in shchwartz space, Schwartz space is closed under all the operations of Navier stokes. There is no way to evolve the field under Navier stokes that can make it leave Schwartz space. Therefore it remains globally smooth for ever and ever. The seminorms do stay bounded but I think that’s beyond what’s even necessary really

[buoyant pike]

Well doesn't your argument that solutions remain in schwartz space apply even without diffusion

[tired hollow]

I think the diffusion is part of how and why they remain in Schwartz is more accurate

Or maybe even more accurate yet is that because they’re in Schwartz space, there is sufficient diffusion

But yes ChatGPT just agreed the Schwartz argument is enough. We showed the seminorms don’t blow up.. but it would be enough just to show membership of initial data in Schwartz and Schwartz closure under Navier stokes. We showed seminorms don’t blow up but you’re right we technically did not need to

[astral vine]

Ah Yes if Chat-Superior Math-GPT says so

(you can make current Chat GPT version fail 1st year problems)

[tired hollow]

I can fail some first year problems too and I’m pretty good at math

[astral vine]

Sure

[tired hollow]

Lol good attitude man

You think your Gauss or something?

[astral vine]

Absolutely not

[tired hollow]

I’m not claiming to be either.. but I do math. And I do it pretty well.. and you don’t know me

I don’t just blindly trust everything ChatGPT says

[lilac barn]

I had a bit more of a look on "your" paper and there are multiple issues:

1. Theorem 1 is nothing but a trivial exercise students prove as soon as they learn about Schwartz functions.
2. The same goes for 2, heat semigroup preserving Schwartz is just a one-liner, doesn't deserve a section on its own.
3. Why does Leray projector maps Schwartz into Schwartz?
4. Much more destructively, your section 5 is purely wrong. You claim that you're able to obtain the bounds because p can be written as u otimes u but that's not true. Pressure is expressed as a particular singular integral of u and it's not at all obvious that you can have Schwattz control on it. Nevermind the fact you include the initial data out of nowhere.
5. Even if your inequality is correct, Gronwall doesn't give you Global control, only local in time as the initial data isnt time-integrable.

[tired hollow]

I collaborate with it. I make it better and it makes me better

[astral vine]

Point 3. is actually false tho

I can produce a Schwartz fucntion s.t. the Leray proj. is not Schwartz

[lilac barn]

Yes, indeed

[tired hollow]

Theorem 1 is not trivial. It shows all initial data lives in Schwartz

[lilac barn]

All initial data?

[tired hollow]

How?

[lilac barn]

More simply, the claim reduces down to saying dx (dxx)-1 dx maps Schwartz into Schwattz which is false. Try picking a bump function and note that the inverse Laplacian can allow you to add any affine function

[astral vine]

Because the leray projection somehow "blows" on low frequencies

with what cocat said, take a Schwartz function with non-0 mean value

you will have some surprises

[lilac barn]

Also even if we grant you the last inequality, that still doesn't give you Global

Gronwall here only gives you local

As the initial data is time independent, you usually need some sort of time integrability of a quantity to get global control.

[astral vine]

Except if you have L1(L^infty) control of the curl of your velocity

so that you would be able to propagate time of existence

Or some other similar Baele-Kato-Majda crietrion

[lilac barn]

Put simply, AI isn't gonna solve something in 3 pages on which mathematicians have been stuck for ages.

[tired hollow]

Initial data is divergence free. Mean is 0

[buoyant pike]

Divergence free functions can have non-zero mean

[astral vine]

Not if smooth he is right

Divergence free L1 is included in mean-0 component wise

[lilac barn]

What? Doesn't matter if it's smooth. Passing to frequency, it simply says (sum xi1) uhat(xi)=0. Since mean is just uhat(0) so you can definitely have non-zero mean without requiring smoothness

[astral vine]

.

On Lp mean value over the whole space makes no sense the integral is not a linear form

[lilac barn]

Idk what you're saying but I'll take your word for it.

[astral vine]

also on other function spaces that do not embbed in L1

[astral vine]

in dimension 1 the issue is mean 0, in fimension higher than 2 this is Hell.

Litteral Hell

[tired hollow]

The key is initial data in Schwartz. That means it decays extremely rapidly in all directions, which forces the mean integral to converge to 0 over all space.

Which makes the Leray projection in Schwartz

[astral vine]

What ever this might be true for the formula obtianed through DuxtDu to compute p

This is where is the issue

but what ever happens you won't sell it to me, and you know what ? That's not important. Submit your work to a reasonable journal, and give talks about it. If it convinces other people and they are happy of your work, this is the most relevant

Who cares about Cocat, Ange' or me.

[tired hollow]

I’m not submitting to journal #justiceforperelman.. no peer review is fucked because people act just like youre treating me. Closed minded.

[astral vine]

I already missed the point on some papers, wouldn't be a first.

[unborn quiver]

I mean, can't you take a divergence free RBF/gaussian thing or something with nonzero mean
I'm probably wrong actually

But regardless, as people have mentioned, there are people trying to remain level headed in trying to understand what you are trying to say. However, you are also kind of applying the faith of "I'm pretty sure this is right" after 'discussing' with chatgpt or whatever. People will be skeptical, but you can reduce skepticism if the paper has bonafide details beyond blind statements saying on the lines of "it's possible to do this"

I'm assuming you've read papers on navier stokes or at the very least have read books on Fourier analysis and distribution theory (considering you are writing on Schwartz functions) so a good place to start understanding the rigor necessary to get a point across is there

[meager dune]

Not sure it is really fair to call people here closed minded lol

[tired hollow]

No I haven’t read books. I’ve talked to ChatGPT and googled and read definitions from wolfram Alpha and then cross referenced with a different AI, Grok, and went back and forth over and over again until we all agreed the proof was correct. I have a bachelors, have two papers self published, one of which was picked up and put on the OEIS. I failed out the first semester of my PhD due to illness.. and that’s where I’m coming from. A solid foundation in mathematics but not an expert by any means on PDEs or Fourier transforms… but maybe that’s part of what helped me digest the full scope of the problem. I wasn’t bogged down in the details of why it hasn’t been solved yet

Oh and I read the official problem statement

[unborn quiver]

That's not really what I was saying, it was more about using other books/papers to determine how to communicate your results in a way that other experts in the field can understand while also being able to give you more direct advice instead of blatant skepticism

[tired hollow]

Ah ok I see thank you

[unborn quiver]

It's a little bit counterintuitive, but a lot of the interesting math would be coming from the details on how you're able to arrive to the bounds you claim to be able to get to based on the structure of the PDE

[tired hollow]

It’s more about the structure of the field under the operations in the PDE

Is that what you mean?

[meager dune]

some other comments: i think it is worth focusing much more on the harder / newer ideas and really drawing those out - this seems a bit like the opposite. like you could just skip over the standard "theorem 1" and stuff and just write a whole document on the seemingly more dubious section 5, and that would be more fruitful i believe

[tired hollow]

Section 1 is the key insight - it’s the one that says all valid initial conditions live in Schwartz space

Theorem 1*

[meager dune]

Well as cocat said it is a standard exercise. i'm pretty sure N-S is often stated just for Schwartz data anyway

[tired hollow]

Also thank you everyone for your feedback even if slightly adversarial at times

[unborn quiver]

What I'm saying is that basically all of the interesting math will be located here, but there aren't any specific ideas/details conveying how these bounds arise

[meager dune]

this is what i mean too

[median forum]

Agree

[meager dune]

I think it is important for the health of maths to be a bit adversarial about things like this

but also to be charitable and definitely not personal

[quick pagoda]

I’m not convinced these bounds are true, e.g., and there’s certainly no argument verifying it here

[unborn quiver]

Even stating a concept such as "... Negatively due to laplacian smoothing" should be supported with a citation or a proof

[unborn quiver]

Yeah exactly

[meager dune]

I feel like a comment in this vein is i would recommend idk stating this stuff a bit more humbly, like the fact it keeps mentioning this is a clay institute prize and states the first result as a "Theorem" and has no citations or anything will indubitably rub people the wrong way

[quick pagoda]

And, like, showing the Leray thing spits out something Schwartz sounds pretty contentious with the above discussion

[tired hollow]

Ok so generally the structure of the proof makes sense but I need to expand on how much of the math I actually show? Totally doable

[quick pagoda]

I’m not convinced the claims make sense at all, but yes actually show some work

Otherwise it’s not a proof but an unsubstantiated claim

[meager dune]

this reminds me that there is a clay research conference in october

[unborn quiver]

Yes, but you should also proactively verify if your work is logically sound as you go. It's entirely possible that this approach will not arrive to an affirmative result bc one of the above sentences is actually significantly harder to prove

Let alone incorrect

But that's kinda just how math is bc your first or second idea isn't always correct... And also why this is still an open problem

[quaint pumice]

this discord continues to make me regret working in ML lmao

[dusk dove]

Why?

[lilac barn]

Can we not continue to have this discussion in adv-pde? The refutations of the above "paper" has been posted and debated quite a bit now. This is now turning into borderline spam.

[meager dune]

Make a thread

[tired hollow]

There hasn’t been a single “refutation” there has been questions and concerns asked and addressed. What’s a better forum to discuss?? If anyone else has any work they’d like to discuss I’d be happy to discuss here in advanced PDEs

[unborn quiver]

I apologize. I was hoping that the provided "work" wasn't going to be AI generated, and the person making it would put in slightly more work looking at other resources that aren't chatgpt or problem statements

[tired hollow]

The nice thing about math is that it doesn’t matter where it comes from if it’s right

And a lot of the “work” does come from me, just FYI

Also I did just read a textbook about Schwartz functions and the math still checks out

Initial conditions live in Schwartz, Schwartz is closed under all operations of navier stokes, therefore a solution exists in Schwartz space and it is necessarily globally smooth

All the math about seminorms being bounded is extra

[median breach]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[tired hollow]

But as a human, I can use AI to help me, and then I can use that knowledge to help someone else. Using AI well doesn’t mean just blindly accepting whatever it says but its true benefit is in learning from what it generates using your own human discernment

[cunning sable]

Ironic, seeing as you seem happy with blindly accepting and posting an obviously faulty AI-made "proof" while barely understanding the basics of the objects you're working with. Since you seem to be adamant in posting work that is not your own I'm removing you from the advanced channels

Please stay on topic

[rustic oak]

That works too

[meager dune]

Wow academia being closed-minded as usual

[quaint herald]

#justiceforperelman

[buoyant pike]

woog they are gone

[worn badger]

Oh , deleted then

[cunning sable]

actually that was a very good message regarding why exactly LLM usage is frowned upon here and what limitations these models have currently, even if a bit off-topic

(where by "LLM usage" I really mean copy-pasting its output or just uploading some pdf file without review or when you are simply not qualified enough to understand and critique its output)

[frank prism]

Not to drift off topic (maybe we can move to #advanced-lounge or something, ping me there if you reply), but I think Tao was dead wrong here.

LLMs have certain strengths (namely for me, the complete recall of all the things in the field, there are many situations where there might be 6 relevant techniques to what I'm doing and I only remember 4 or 5 off the top of my head, but the LLM remembers all 6) but are so terrible at actually being correct, that I don't think it's a good assessment, or even really a fair comparison: do we want LLMs replacing grad students, as opposed to developing in the future into hopefully useful research tools to help mathematicians?

[worn badger]

Ok I can put it back here

The problem arises when the human corrects the AI only about topics they know enough about, when discussing topics at the edge of their understanding capacity. Eventually the LLM learns (in-context) to create unfalsifiable narratives/code/etc because it's easier than creating something of value. This is a really big open problem in AI right now, with no solution in sight for at least the next couple years. Until then it's going to remain frowned on when relied upon in contexts where consistency and verification are core expectations of participants involved, such as here on this server.

If anyone is interested in why exactly that happens or what the precise open problem statement or formulation is we can chat somewhere else 😌

[regal patio]

i'm guessing this means nobody wants to see my automated theorem prover demo(s) hosted for free on h200s (using specialized llms)

[buoyant pike]

[regal patio]

honestly i find it's like magic, but at the end of the day it's just math and a ton of lean4 ^^

[cunning sable]

not here -- we have a #computing-software channel where Lean and automated theorem proving would be on-topic

[regal patio]

ooh thank you kindly for this

[arctic whale]

I am looking at the proof if this thm

I don't really get this step

How did they find this sequence?

They say it just follows from 3.5.5 (that's just the fact that L^2 norm of beta(x, t) is non-increasing as a function of t

might be obvious but I can't see how that just follows

[lilac barn]

What's E

[arctic whale]

Energy of the form

[median forum]

You have this formula for energy, and the energy is decreasing and bounded above

[wooden mango]

I was reading something about separation of variables, and how Stone-Weierstrass allows us to conclude that the algebra generated by solutions obtained by separation of variables are dense in the solution space. How do we conclude this?

[median forum]

You just show that you can separate points and then you can apply the theorem

[rotund jetty]

Stone Weierstrass is one way to show that the Fourier basis is dense in $L^2$, which therefore motivates looking for a series solution to the PDE. But I don't see how it motivates separation of variables. Another motivation is that if you take the span of product functions of two function classes that individually have dense span, then the product functions have dense span.

[untold delta]

L

[wooden mango]

I agree with this. The solutions form a vector space, and thus an algebra. Stone-Weierstrass basically gives conditions for the existence of a dense subalgebra, and if we can show that this dense subalgebra is generated by product functions obtained by separation of variables, we are done. My only qualm with this is that I am not sure how we should apply this. Your remark that a product of dense subsets is dense in the product set is fair, but I don't see why this is helpful here

The other thing is that the domain for our functions should be $H^2_0$ (or something very similar), not $C_0$, but $H^2_0\subset C_0$, and the topology of $H^2_0$ is stronger than that of $C_0$

[untold delta]

Sleepybear

[undone sable]

I think the closest rigorous thing to what the book is saying is we take the eigenfunction expansion of the solution $f(t, x) = \sum_{j = 1}^{\infty}a_j(t)u_j(x)$ and then solve for the coefficients $a_j(t)$.

[untold delta]

L

[undone sable]

This motivates the study of the eigenfunction expansion, since it immediately gives us the solution $f$

[untold delta]

L

[minor mulch]

he didnt say the product of dense sets is dense

maybe you can do it for C^2 functions

and then try to extend to H^2

for C^2 functions you can use stone weierstrass to approximate the 2nd derivative

and then hopefully integrate that to control the lower order derivatives

maybe assuming things are compactly supported and then removing that via another approximation later

[undone sable]

the remark about density in the product set shows that to get a basis eigenfunctions on a product of surfaces $M \times N$, you only need to get bases on $M$ and $N$ separately, then take all the products

[untold delta]

L

[undone sable]

I mean, this works because the laplace operator acts nicely on products

[proven compass]

How could I possible generalize this result to $\mathbb{R}^k$, the ODE: $\frac{d^n}{dx^n}y= \text{a power-tower composed of n y's}$ has the solution $y(x)=\text{slog}{b}(x)=\text{ln}\left(-\frac{1}{2\pi i}\displaystyle\oint{\gamma} \frac{b^z}{b^z-x}dz\right)$ where $\text{slog}b (x)$ is the base $b$ Super Logarithm of $x$. If we call the power-tower of $n$ $y$'s: $ P_n (y)$ then my guess is that we would have to find some function $L(f)$ that turns are PDE from $F=0$ to $L(F)=\displaystyle\sum{i=1}^{k} P_n (F_i)=0$ where $F_k$ is the $k$th component of a function living in $\mathbb{R}^k$

[untold delta]

StellerYak

[buoyant pike]

[astral vine]

Yeah you are right. Th actual good motivation is Spectral Theorem on Hilbert spaces and the Fredholm alternative for (unbounded, essentially) self-adjoint operators with compact resolvents.

[primal gate]

hey guys, can I ask about Lyapunov stability here?

[buoyant pike]

Well, we certainly can't stop you

#dynamical-systems might be a better place for it though

[primal gate]

ok thanks

[lilac barn]

Navier-Stokes gets solved again: https://arxiv.org/pdf/2507.18063

[buoyant pike]

https://arxiv.org/abs/2507.18063

[meager dune]

As a non-expert lol: how does this paper look lol

[astral vine]

Roughly speaking the idea of the paper is to reduce the Analysis to a perturbed heat equation (with some additional parameter lambda) called the Lamé evolution problem (which is equivalent somehow to the non-linear heat equation by itself, for fixed parameter, up to an appropriate rescaling).

For such a perturbed Heat equation equation the author claims that his non-linear heat equation admits unique smooth Global-in-time solutions, with some bound uniform with respect to lambda.

The authors seems to claim that sending lambda to infinity allows to construct a limit solution to Navier-Stokes and due to the uniform bound and smoothness properties of the approximating problem.

[quick pagoda]

It’s a much more real attack than some recent ones

[astral vine]

The issue to me is about claiming existence and uniqueness for such a perturbed Heat equation

this is where there is a hole according to me

Moreover, there are a lot of inconsistencies and weird things, like writing the symbol of the differential operator

[untold delta]

Functionanatolysis

[astral vine]

as

[untold delta]

Functionanatolysis

[astral vine]

which is not true

PseudoDifferential clauclus does not work like this

[quick pagoda]

I’m not convinced this is right but it’s, like, a way more sophisticated one

So it’s better than AI ones immediately

[astral vine]

not so sure it is not AI generated I would say

[quick pagoda]

That would be quite the twist, it looks better

[median forum]

Author does seem to have other papers about this Lamé equation, but not really others about N-S which is a bit strange.

[astral vine]

The Lamé system arise quite naturally when studying viscous fluid flows, so by itself this is not really weird

[meager dune]

Thanks

[buoyant pike]

Greetings and salutations

I am trying to figure out what the Green's function for the biharmonic equation on the sphere is

So $\Delta^2\phi=f$ on $S^2$, the unit sphere

[untold delta]

Angetenar

[buoyant pike]

For reference, the Green's function for the Poisson equation, $-\Delta \phi=f$ is $G(\mathbf{x},\mathbf{y})=-\frac{1}{4\pi}\log(1-\mathbf{x}\cdot\mathbf{y})$

[untold delta]

Angetenar

[buoyant pike]

One way of deriving this is by looking at a spherical harmonic series

So for the Poisson equation, $G(\mathbf{x},\mathbf{y})=\sum_{n=1}^{\infty}\frac{2n+1}{4\pi}\frac{1}{-n(n+1)}P_n(\mathbf{x}\cdot\mathbf{y})$

[untold delta]

Angetenar

[buoyant pike]

Coming from the Convolution theorem and the fact that Laplacian of Y_l^m is -l(l+1)Y_l^m

Ok so for the biharmonic equation, the bilplacian of Y_l^m is l^2(l+1)^2Y_l^m

So the corresponding Green's function should be $G(\mathbf{x},\mathbf{y})=\frac{1}{4\pi}\sum_{n=1}^{\infty}\frac{2n+1}{n^2(n+1)^2}P_n(\mathbf{x}\cdot\mathbf{y})$

[untold delta]

Angetenar

[buoyant pike]

Which is all well and good, if you compute partial sums they look like this

Ok

So for a closed form

Online, you will find several sources which present a closed form of $G(\mathbf{x},\mathbf{y})=\frac{1}{4\pi}\mathrm{dilog}(1/2(1-\mathbf{x}\cdot\mathbf{y}))$

[untold delta]

Angetenar

[buoyant pike]

Which looks like

It is reflected across the y axis!

Changing it to 1/2(1+x*y) fixes this

So what's going on

[lilac barn]

Have you checked https://arxiv.org/pdf/2507.03455v1

[waxen bobcat]

That's quite recent

[buoyant pike]

All this wonderful machinery and no closed form expression for S^2

[broken hamlet]

Does anyone know of a more modern reference for this inequality? This comes from Hamilton's book "Harmonic maps of manifold with boundary"

$L^p_k = W^{k, p}$ in the analyst's notation

[untold delta]

shingtaklam1324

[rotund jetty]

Maybe M. Taylor's books?

[broken hamlet]

yup, it's Ch 15.2 in Taylor (Book III), at least for the interior estimates. Thanks!

[arctic whale]

In PDEs and geometric analysis, many proofs go like "consider the following function (slams a very wired cooked-up function)" and then proceed to show how the maximum principle (or some form of a comparison argument) will give you what you want.
In some instances the function is not convoluted enough that you can kind of heuristically justify why would've someone though about that. In some other instances you can kind of reverse-engineer it if you know what this function needs to satisfy and you describe that as an ODE that you can solve. But in many other instances, the given function just comes out of nowhere. How should I approach that? I am trying to learn as much as I can from proofs but when the proof boils down to "behold the magical function", I am not sure what I should learn from that.

[heady silo]

what is an example of a magical function? maybe we can better answer your question if you show us an example 😄

[quick pagoda]

I mean, as you said, sometimes you can work backwards

“Hey, I think this statement is true, but if I was to show it by producing an example in this fashion, what would it have to look like?”

“If I could solve problem P with f such that XYZ holds, what must be true of f”

Or, what are sufficient condition, and then cooking up f to satisfy those etc

[verbal nebula]

Anyone have a good reference for Paradifferential Calculus? My advisor was mentioning some work by Chemin on this, but he seems to have forgotten the title

[turbid scaffold]

\sout{Does anybody here know about disconjugate odes?}

[broken hamlet]

So I think the question has two parts. First is what properties this special function should satisfy (eg. some growth/decay condition, regularity, ...). The second one is how to construct a function that has those properties. I think the first question can usually be answered by looking at the proof, but the second one mostly comes from trial and error (and experience makes this process faster)

[undone sable]

They are covered in volume 3 of M. Taylor's PDE books.

[verbal nebula]

Damn Volume 3

[lime crow]

Can anyone recommend an introductory PDE book with decent theory and good collection of problems?

[jagged adder]

Strauss was good

You can supplement a lot of the theory with Evan’s once you’re more familiar with the content