advanced-pdes

[nova vault]

consider $u_t+\partial_x f(u)=0, \quad(x, t) \in \mathbb{R} \times(0, \infty)$

[untold delta]

ben

[nova vault]

this wil be a while because i need to rigorosly clarify the rankine condition

with $f \in C^2(\mathbb{R})$ and initial data $u(x, 0)=u_0(x) \in L^{\infty}(\mathbb{R})$

[untold delta]

ben

[nova vault]

then what is the issue @unkempt wharf

i've stopped typesetting, waiting fr the exact confusion

hope this helps f′(u)=v(u)+uv′(u) is the character speed

where v = f / u

and this is same as v when f = C u

btw there are rigorous proofs fr these, as you prolly know

in diff geom sense

i only used the wordline thing to call a param curve because i hv a string theory back

[neat spoke]

Are fractional sobolev spaces related at all to fractional derivatives in the sense of this article: https://en.m.wikipedia.org/wiki/Fractional_calculus ?

[nova vault]

@neat spoke can you be slightly mr sepcific

like there are cnnection bw frac sololev are where the riez derivs are

but then are lioville derivs which arent the same

if you're more precise i can help

[quick pagoda]

I mean, the H^s Bessel potential-y spaces are kinda exactly designed to work with the Sobolev multiplier-y fractional derivatives (with maybe mildly sussy details for 0<s<1 and bad domains)

Since it’s like the space where (1+Delta)^s/2 is nice

[astral vine]

it is never "truly defined" as such on domains even smooth, because it involves compatibility boundary conditions. Only holds for R^n and manifolds without boundary

[quick pagoda]

Well yeah but nothing can be nice on domains

[astral vine]

It can but not that way

[quick pagoda]

Fair

Boundary conditions are the bane of my existence

[astral vine]

concerning this wikipedia article with such definition it necessarily restrict to the analysis on the half-line [0,+ infty), or [0,T], T<\infty.

[untold delta]

Functionanatolysis

[astral vine]

note that the restriction s in [0,1/p) is mandatory, in general it does not hold as such for say s in (1/p,1]

However, one can define such fractional derivative on the whole line instead of the half-line, for which I won't give details here

[untold delta]

Functionanatolysis

[neat spoke]

What is meant by the $\simeq_{p,s,n}$ notation? That the two norms are equivalent?

[untold delta]

MisterSystem

[median forum]

Yeah $A \simeq_{p,s,n} B$ means $A < C_{p,s,n} B, B < C'_{p,s,n}A$

[untold delta]

zeke

[astral vine]

Zeke did answer well. Note that on the whole space case the restriction on s has been removed, but CANNOT be removed on the half-line.

Finally proving this kind of result is highly non trivial and requires deep (but very well known) harmonic analysis.

And on the whole space case the definition IS NOT the same has the one in your Wikipedia article (but still has a form close to this one)

[neat spoke]

I assume it's because on the whole space you can apply a dilation argument?

Like, how you guess correctly for which q you have an embedding W^1,^p(R^n) -> L^q(R^n)

Maybe this is not the issue here because R_+ is closed under scaling by posite reals

[astral vine]

Not really

[neat spoke]

It's just the first thing that comes to mind nowadays when thinking about proving embedding results

[astral vine]

Guess for free it holds for p=q

Then dilation argument

It holds q=p*

Then it holds for q between p and p*

[neat spoke]

Isn't this kind of counterintuitive?

I would guess that the bigger the k, the bigger the subset (in terms of dimension) that could serve as a domain to evaluate f.

Because the bigger the k, the more regularity you have.

[upper knot]

I got a question

When we're trying to solve the poisson equation for a C2 function u in a domain U, we derive a representation formula for u using the fundamental solution for the Laplace equation.

Now we do this via applying the Green's formula on U minus a ball.

My question is, when we apply the Green's formula directly on U and not U minus a ball, we do not get an additional "u" term, which we do get after applying green on U minus a ball and taking the radius of the ball going to 0. Why is that?

I mean, can we apply the Green's formula directly on U anyways? It does have C1 boundary

Is it because we're correcting for the point where the fundamental solution of the Laplace equation would not be defined? If so, then the integral should be over U minus that point, not U, right (I mean, even if it is, it doesn't change anything really, right? Coz like the integral doesn't change over zero sets? Am I going right or am I confusing stuff?)

But even after correcting for it, why couldn't I just apply Green's formula on U minus the point? I would still not get the additional "u" term? Why does it only come after working with U minus a ball and then approximating the point by the ball?

Maybe the problem could be that I cannot apply green directly on U minus the point, but I dont know why.

[astral vine]

Check just with k=1.

And then k=n

[digital ibex]

This is kinda a follow up to my question yesterday, can someone please tell me what’s being “added and subtracted” here?

Here is the full page for context:

(Im looking over lecture notes, not a textbook, that’s why there aren’t equation numbers)

[astral vine]

Wdym by "what's being added and substracted"

[buoyant pike]

So you have [\tau_{ij}=\frac{c_1}{2}\left(\pdv{u_i}{x_j}+\pdv{u_j}{x_i}\right)+c_2(\nabla\cdot\mathbf{u})\delta_{ij}] and you write $\mu=\frac{c_1}{2}$ and then you split [c_2(\nabla\cdot\mathbf{u})\delta_{ij}=-\frac{2}{3}\mu u_{k,k}\delta_{ij}+\lambda u_{k,k}\delta_{ij}] so $c_2=-\frac{2}{3}\mu+\lambda=-\frac{c_1}{3}+\lambda$ so $\lambda=c_2+\frac{c_1}{3}$

[untold delta]

守沢 千秋

[digital ibex]

So we’re defining the c_2 div(u) term in terms of mu?

$c_2(\nabla \cdot{\mathbf{u}}) := -\frac{2}{3}\mu u_{kk}\delta_{ij} + \lambda u_{kk} \delta_{ij}$?

[buoyant pike]

What do you mean

[untold delta]

KySquared

[buoyant pike]

The c2 div u term is already in tau ij

[digital ibex]

Right
But what “split” is going on? The div?

[buoyant pike]

Yes

[digital ibex]

Ohhh I think I see it

Lemme just write this out so im sure

[digital ibex]

I think(?) they should be matrices as the i’s and j’s from the identity dont get contracted w/ the k’s in u

But i can easily see how these could also be vectors since vI = Iv = v

[buoyant pike]

I have no clue what you've written

[digital ibex]

$\mathbf{u} = [u,v,w]$, so $\nabla\cdot \mathbf{u} = \partial_x u + \partial_y v + \partial_z w$

[untold delta]

KySquared

[digital ibex]

…
How on EARTH did i switch div with grad in my head when I got to the next step

$\mathbf{u} = [u,v,w]$, so $\nabla\cdot \mathbf{u} = \partial_x u + \partial_y v + \partial_z w$\
\begin{align}
c_2 \left(\nabla\cdot \mathbf{u}\right) \delta_{ij} &= c_2 \left(\partial_x u + \partial_y v + \partial_z w\right) \delta_{ij}\
&= \left(\lambda -\frac{2}{3} \mu\right)\left( u + v + w\right)
\end{align}

[untold delta]

KySquared

[digital ibex]

?

[buoyant pike]

?

Where did the derivatives go

[digital ibex]

Oh
When i did this I was thinking if the div was some scalar multiple of the field’s sum, guess that’s wrong

Im just gonna go back to square 1 so i stop looking like a fool

[quick pagoda]

I mean, if it makes sense for k=1, iterating that should be pretty similar

[arctic whale]

Anyone familiar with this proof from Jost? I don't see how the energy of b_{\lambda,t} is related to the d^2 between two lifts of c_{\lambda}. Also how is it related to the L2 distance between g and g_{\lambda}?

I am talking about the non-simply-connected case of course.

[buoyant pike]

Probably better to ask this in #diff-geo-diff-top

[arctic whale]

I feel like in this server, PDEs ppl are more likely to have read Jost than diffgeo ppl.

[mortal slate]

Is there a flow which preserves the uniform lower bound on a manifold with say sectional curvature >= c

Say a Euclidean submanifold

[silk grail]

Hi, I am reading Hormander's first volume on Linear partial differential equation. In this proof I am not able to see where are they using the fact that (y_j \not\in f(V)), can someone please help me with this. Thanks a lot.

[untold delta]

Keshav

[quick pagoda]

Since the things are supported off the image of V under f, you’d have the v terms all equal to the u_eps terms

And those terms don’t go to zero

Whereas the v ones get some cancellation-y stuff going to 0

[silk grail]

You mean, under f they are same i.e. v \circ f = u_epsilon \circ f...

[quick pagoda]

Yeah

That’s where it’s “used” ig

[silk grail]

I see, did not notice that earlier... I was thinking it was used in construction somehow... but now I see its fact its equal to u_epsilon \circ f there its used. Thanks a lot...

[worn chasm]

Hey,

I'm trying to solve $-\nabla^2u=f$ with $\left.u\right|_{\partial\Omega}=g$ on a cube $[0,\pi]^3$ but i'm somehow struggling

I split $u=v+w$ such that $v=0$ on $\partial\Omega$ and $w=g$ on $\partial\Omega$. It gives me the now homogeneous problem $-\nabla^2v=f+\nabla^2w$ with $v=0$ on $\partial\Omega$ which is not so hard to solve analytically.

But then comes the coarser part, finding $w$. I thought of defining $w$ on every cube's face and them summing the different contributions. Each contribution is such that it is evaluated at $g$ on its face and $0$ on the opposite face.

The general form of the $x=0$ contribution (for example) is defined as $$W^{(x=0)}(y,z)=\sum_{p,q\ge1}a_{pq}X(x)\sin(py)\sin(qz)$$ with $a_{pq}=\frac4{\pi^2}\iint_0^\pi g(0,y,z)\sin(py)\sin(qz)\mathrm dy\mathrm dz$

Processing $X$ i find it to be $$X(x)=\frac{\sinh(r(\pi-x))}{\sinh(r\pi)}$$ where $r=\sqrt{p^2+q^2}$

I can do that for every face and sum up the contributions. Another problem arises, on the edges i have twice the contribution and even 3 times on corners.

Is it one right way to obtain a pure analytical solution with no prior knowledge on $g$ or am I in the wrong ?

[untold delta]

Hokkaydo

[astral vine]

on a cube it is not possible to construct such a w in general. There are deep issue with the trace theory and the ability on lifting trace, due to the corners

especially if g is not periodic

The 3-dimensional Torus IS NOT equivalent to the cube in terms of regularity theory for PDEs

people did wrote a lot of bullshit in the literature concerning this fact

For instance the best you can have for solvability of the bilaplacian with 0-boundary condition and RHS in L², on the cube is u in H^{7/2}.

while on the torus it is H^4

People should stop consider cubes/square/hypercubes as the torus. both theory coincide only in the case of an interval

So just stop pls

Teachers/Supervisors do have bloods on their hands

[quick pagoda]

This is very curious and still has the point of nasty differences, but I would wonder if that nabla^2 is meant to be the laplacian that is not bi

[worn chasm]

Alright thanks aha
I figured another approach, i you have some spare time, maybe could you validate it ?

[worn chasm]

Yeah, i mean $\nabla^2=\frac{\partial^2}{\partial x^2}$

[untold delta]

Hokkaydo

[buoyant pike]

Fix your latex

[astral vine]

Oups my bad, I fully brain farted. Lmao

[astral vine]

For the Laplacian itself, yet it is not exactly the same to consider the bidimensional Torus or the square.

[quick pagoda]

Oh yeah I wasn’t saying they’re the same there, but that it’s a good point and that I think it was a different operator

[nova vault]

where did you get this from btw ? ref?

[astral vine]

Cannot right now, ping me in exactly 24h if I forget

[grave matrix]

@astral vine

[buoyant pike]

9 minutes early

[austere raft]

If f: R x R^n -> R^n is periodic on t (f(t+w,x) = f(t,x)), then the solution 0 is stable iff it is uniformly stable. Does any one have a reference for this result?

[austere raft]

We are using lyapunov stability here

[tired hollow]

Anyone of y'all well versed in partial integro differical equations?

Dealing with heat memory equations rn

Something doesn't add up

Ima crash out
Could use any tips to make my life easier with them accursed inverse Laplace trasfroms like wth
I know analytical methods don't always work and its compelling to make use of numerical analysis sometimes but i just have a feeling that...my analytical part could use some work..

[buoyant pike]

What is your actual question

[tired hollow]

So I'm having trouble with the inverse laplacian trasform

It's just pissing me off

I don't think I'm missing out on any theory, i was just wondering if there is a particular way to think about them that I'm missing

Like you know? Certain things seem easier when you think in the right direction

I was asking that

For inverse laplacian transforms
It doesn't haven't to be in the context for IPDEs because the mechanism of the inverse laplace transform is identical in any context
Like have you seen those questions with wierd rational approximations because the denominator is a absolute nightmare

[alpine carbon]

Hi. Can anyone give me a hint as to how to approach part b) of this question?

In this case, $\delta$ refers to the dirac delta function.

[untold delta]

Good Luck HSC Students!

[median forum]

Use change of variables

[alpine carbon]

How so? Like x = f(u)?

[median forum]

yeah

[austere raft]

If f: R x R^n -> R^n is periodic on t (f(t+w,x) = f(t,x)), then the solution 0 is stable iff it is uniformly stable. Does any one have a reference for this result?

[astral vine]

What is f, what is your ODE/PDE problem ? Afaik we do not read in minds

@nova vault @grave matrix Theorem 7.2.2.3 from Elliptic Problems in Nonsmooth domains gives that the solution in the square/cube/hypercube for the bi-Laplacian is NOT H^4, and gives exactly why.

The precise statement concerning the fractional regularity is more involved and combines the results of several advanced papers

[austere raft]

x' = f(t,x), we are assuming f is continuous and the solutions are unique

[astral vine]

I think this is not true and there are counter examples

Unless I am missing something

[austere raft]

Okay, can you point me to any counterexample? This statement is in my book and I didn't agree with an step of the proof

This is from Hale

[astral vine]

I am looking for it

Okay my bad the coutner examples I had in mind were for something close but different

I apologize

(it was still for ODEs, and closely related, but tied to the linearized vs non-linear behavior (a bit more than this but still))

[austere raft]

[quick pagoda]

Interesting

[copper finch]

do you guys have tips on learning pde stuff? I am recently learning basic energy methods and lax milgram stuff for sobolev spaces (poisson equation) but am kind of new to pdes

[nova vault]

thx fr ref

[alpine carbon]

Sorry for the late reply. I thought about it for a while, and I think I got somewhere. If you have the time, could you look at my reasoning?
[ \int_{-\infty}^{\infty}\delta\left(x-a\right)dx=\int_{-\infty}^{\infty}\delta\left(f\left(u\right)\right)f'\left(u\right)du ] We use the substitution $x - a = f(u)$, just because if $x=a$ then we can set $u=a$ too. \

Then, because the delta function only contributes at the root of $f$, we can evaluate $f'$ at $a$;
[ \int_{-\infty}^{\infty}\delta\left(x-a\right)dx=\left|f'\left(a\right)\right|\int_{-\infty}^{\infty}\delta\left(f\left(u\right)\right)du ]
We have the absolute value sign so we don't "swap" the integral bounds. \

Then, we can remove the integral sign because of the properties of the delta function (not too sure on this).
[ \delta\left(x-a\right)=\left|f'\left(a\right)\right|\delta\left(f\left(x\right)\right) ]
As required???

[untold delta]

Good Luck HSC Students!

[alpine carbon]

Reposting the question:
https://media.discordapp.net/attachments/908110296239980564/1428491279058079794/Screenshot_2025-10-17_at_8.14.10_am.png?ex=68f4abed&is=68f35a6d&hm=6364494a70ff6eedcce8f0c820222246341ff68b262e56745db42019bb0e350f&=&format=webp&quality=lossless&width=2473&height=436

[alpine carbon]

"because of the properties of the delta function"
Kinda made up this sentence, but I think as long as the delta function on the left and right hand side has the same 'root' (i.e in this case, x = a), then it should be fine, right?

[rotund jetty]

to do this rigorously, you need to pair $\delta(f(x))$ with a test function $\phi$ and show that it equals the pairing of $\frac{\delta(x - a)}{|f'(a)|}$ with $\phi$

[untold delta]

L

[rotund jetty]

The main challenge here is how to even compute the pairing $\langle \delta(f(x)), \phi(x)\rangle$

[untold delta]

L

[rotund jetty]

Because composition with a smooth function $f$ is only defined when $f$ is invertible with smooth inverse.

[untold delta]

L

[rotund jetty]

Here you need to sort of isolate a neighborhood around $a$ and then use linearity of the pairing and the usual definition for invertible f

[untold delta]

L

[alpine carbon]

How do I isolate a neighbourhood around a? Also, I'm not sure what the "usual definition" for invertible f refers to.

[alpine carbon]

Computing the pairing... is that figuring out what phi(x) is?

[quaint herald]

It would help if you gave some context for this, i.e. the course you are studying and the way that $\delta$ and operations involving it have been treated in your notes. Some courses treat it pretty informally, so it is not obvious from your question whether you are expected to work with distributions rigorously or play a little fast and loose, like "changing variables" in integrals that aren't actually integrals.

The proper way of making sense of objects like $\delta(f(x))$ is through the notion of the \emph{pullback of a distribution}. You can always pull back distribution by smooth submersions, and for diffeomorphisms, the formula is what one would expect by naively writing the pairing with a test function as an integral and changing variables purely formally.

[untold delta]

grobmez

[alpine carbon]

Sorry for the late reply. The course I'm taking right now is a PDEs course. We had just covered fourier series, heat equation, laplacian PDEs, and now we have been introduced to this topic. The course is based off Olver's "Introduction to Partial Differential Equations" where he talks about generalised functions. What you mention as distributions, smooth submersions and diffeomorphisms are completely foreign to me.

[quaint herald]

Cool, I am glad I asked because indeed this treatment is quite informal. It is a bit hard to prove an identity for $\delta(f(x))$ if one does not first define what this even means. Because of this it is unclear exactly what your prof wants. Avoiding any advanced terminology and concepts and treating $\delta$ informally by writing things as ``integrals", the idea is as follows:

[untold delta]

grobmez

[quaint herald]

If $f:\mathbb{R}\to\mathbb{R}$ is a smooth bijection with smooth inverse, then we can change variables" in the integrals" to write
$$\int_\mathbb{R} \delta(f(x))g(x),dx =\int_{\mathbb{R}} \delta(y) g(f^{-1}(y))\left|\frac{dx}{dy}\right|, dy=\frac{g(f^{-1}(0))}{|f'(f^{-1}(0))|}$$
$$=\int_\mathbb{R} \delta(x-f^{-1}(0))\frac{g(x)}{|f'(f^{-1}(0))|},dx $$
where $g$ is a test function. Examining the terms at the extreme LHS and RHS we see that $$\delta(f(x))=\frac{\delta(x-f^{-1}(0))}{|f'(f^{-1}(0))|}$$ because they produce the same result when ``integrated" against an arbitrary test function $g$ (and indeed this is the correct formula if one approaches these objects a little more rigorously).

The above is still valid if you weaken your assumptions on $f$ a bit, for example as in your question. There it is assumed that $f$ has a unique root and the derivative is nonzero there. The idea is that for $x$ near $f'(0)$, $f$ is \emph{locally} a smooth bijection with smooth inverse, and the part of the first integral in my first equation where $x$ is far from $f^{-1}(0)$ is zero, because $\delta$ is supported at $0$, so you can break your integral into two pieces and argue as before for the piece that is not zero.

[untold delta]

grobmez

[alpine carbon]

Got it. So for now I will just have to keep in mind this idea of 'changing variables' in the integral, and then trying to equate the integrals using delta and various test functions. Thank you!

[rotund obsidian]

Does anyone understand how I obtain the integral of du/dn in the second line ?

[rotund obsidian]

Also why do I have $\int_{\partial B_R}f dS rightarrow 0$ when R goes to infinity ? (And f is integrable)

[untold delta]

Léone

[rotund obsidian]

I solved this question

[worn chasm]

My bad
It rendered well in my editor when I exported the file
Here is the fixed version

[short oar]

I have this proof by induction for an estimation for harmonic functions (evans page 30), but whenever I try insert the k-1 estimate into the k one as done in the proof, I am left the almost the same as in (31) but a factor of k^n/(k-1)^n which is bigger than 1, so a more loose estimate. Am i missing something? How to get rid of it

[short oar]

I think it works if you dont use $||u||{L^1(B(x_0,r))}$ as estimate for k-1, but keep the induction at $|\int{B(x_0,r')} u(x) dx |$ (applied to $r'=\frac{k-1}{k}r$ ) with absolute value outside the integral and then you can use $\int_{B(x_0, \tfrac{k-1}{k}r)} u(x),dx = \left(\tfrac{k-1}{k}\right)^n \int_{B(x_0, r)} u(x),dx$ following from mean-value property

[untold delta]

Knechterich

[sonic olive]

Don’t worry about the constants too much. You can always get them from scaling

The point is you are controlling a higher order derivative by a lower order one

[loud patrol]

should i audit my grad level applied pdes as an undergrad (ive only taken intro analysis) - is this a safe bet? im also planning on taking an analysis and geometry class with a focus on PDEs

[buoyant pike]

Do you have the syllabus

[loud patrol]

yea

the instructor just said it requires intro analysis like the class im taking would be sufficient

but im an undergrad so there might be more hoops for taking it for simultaneous grad and undergrad credit if i want to get a masters at my current institution

otherwise it just appears on my transcript without a grade

[loud patrol]

I can ss

the prof also has this analysis and geometry class

\

he said he can meet in person to discuss but idk what his schedule is

like he said he's supposedly less busy after early october and well i guess we're almost at november

[buoyant pike]

Oh yeah I think this should be doable

[loud patrol]

like as an undergrad

[buoyant pike]

Yeah

[loud patrol]

idk i am also taking quantum 1, classical 1, and physics math methods and possible intermediate abstract alegbra like ringa dn group theory

physics math methods is just applied pdes

so maybe just workload wise it might be better to audit one or the other

[buoyant pike]

Well don't overload yourself

[loud patrol]

out of these two classes I am interested in

like the pde realted ones

[buoyant pike]

Physics math methods is a fake class don't take it

[loud patrol]

real but im doing a physics degre

[buoyant pike]

Oh

[loud patrol]

and the prof is cool

along witha math degree

[buoyant pike]

Well

[median forum]

Having difficulty establishing the bound $\norm{\langle x\rangle^k e^{it\Delta/2} f}{L^2(\R^d)} \leq C \langle t\rangle^k \norm{f}{L^2}$. Not quite sure what tools are needed/what direction to go in.

[untold delta]

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

[loud patrol]

should i just talk in person with the prof teaching with the math pde classes

[quick pagoda]

Ask your school's admin/advisors

[loud patrol]

yea i have an appointment with them i also may meet with the prof too

i need to study my ass off for my coming analysis exam though like

i hope i will be ok if i do

[loud patrol]

for the time being

this is for next sem

[quick pagoda]

The pde course feels a bit more vibes-y, the analysis-geometry one seems nifty

[loud patrol]

idk what is monotonicity like monotone convergence

like strictly increasing or decreasing

omg lol we learned about hugyens principle in physics

well he said the analysis and geomtry course might be more rigorous so why do i not audit that what idk

idk what else about his class structure well idk i might just talk in person

[quick pagoda]

No very much so not

[quick pagoda]

It definitely looks more “real”

[stray fox]

If we do not have the japanese bracket and let k=1, we can consider the fourier transform inside the L^2 norm $|x|^ke^{it\Delta/2}f,$ which gives $|\nabla_\xi|(e^{-it|\xi|^2/2}\hat{f})=e^{-it|\xi|^2/2}|\nabla_\xi|\hat{f}-(it)|\xi|e^{-it|\xi|^2/2}\hat{f}$, which transform backs to $e^{it\Delta/2}|x|f-ite^{it\Delta/2}|\nabla_x|f$. If you are assuming the L^2 norm |x|f and f to be constant, I think this gives the bound. Maybe you can look into this direction.

[untold delta]

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

[median forum]

Yeah sorry the norm on RHS should be the weighted one.

[stray fox]

Sorry i dont quite get it.

[median forum]

Like it should be $\cdots \leq C (\norm{f}{L^2} + \norm{\langle x \rangle^k f}{L^2})$

[untold delta]

zeke

[median forum]

Or something similar

[stray fox]

If you are concerned about the japanese bracket, I think after you fourier transform the left hand side, then it follows from sobolev product rule on the frequency domain. the <t>^k term comes from calculating the quantity $\norm{e^{-it|\xi|^2/2}}{H^k\xi}$

[untold delta]

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

[median forum]

Is this finite?

[stray fox]

obviously not lol mb I think i did things wrong.

[lilac barn]

Doesn't this fail at t =0?

Or is the first norm in t as well?

If the norm on RHS is the weighted one, then it does follow: go on the Fourier side where the multiplier <x>^k is <par_xi>^k which amounts to bounding the 0-th norm and the k-th norm. The 0-th norm is trivially bounded and for the k-th, you can take derivatives on both terms and bound.

[celest dust]

Hi! I have a little doubt. Do someone know in which spaces H^{s,p} (Besse potential spaces, including negative values of s) can i solve the heat equation u_t-\Delta u = partial_j u, with Dirichlet boundary condition, for instance. Thanks!

[loud patrol]

how much computational ode and vector calculus would I need for studying PDEs (applied for physics and theoretical using analysis) i sadly don’t rlly remember much

multi variate/vector calculus

[nova vault]

i know

[celest dust]

And may you tell me?

[nova vault]

bry ws busy tlking on disc

this is r^n @celest dust

?

[celest dust]

For instance. Although I would also be interested in the case of bounded domain

[nova vault]

okay

wait lemme ask before i anser in r^n ya mean ya wabt an snwer in direchlet domains

cuz i was gona talj about semigrps H^p(r^n)

but then since ya want bounded domains; like c^1

it's a diff kinda issue

[astral vine]

For which domain ?

And what does interest you the full Cauchy problem . Or just the initial value one, or just the forcing one ?

What do you aim to have in Hsp, the solution u itself ? The initial data u_0 ?

The answer is certainly yes.

But you need to provide more info about your meaning of "solving"

In general for a bounded Lipschitz domain, I can say that -∆-Dj generates an holomorphic semigroup on Lp for all p between 1 and infinity (included)

It even has the BIP property so that you can identify the domains of the fractional powers of its operator with actual Hsp.

For C¹ domains you can reach the identification more Hsp (a wider range of indices s,p) spaces than for Lipschitz

[rare oasis]

anyone here has experience working numerically with the KP equation? or good literature references?

[buoyant pike]

Kadomtsev–Petviashvili?

[buoyant pike]

Oh also numerics go in #numerical-analysis

[rare oasis]

yes

[buoyant pike]

<@&268886789983436800>

[restive sail]

Let's please keep shitposts out of the advanced channels @remote falcon

Please don't do that again

[sullen ember]

can someone give me a hint regarding this exercise? Applying the general Sobolev inequality first leaves me with powers of |u|, I don't know how to elegantly finish the proof. 2* here is the critical Sobolev exponent 2d/(d-2)

[sonic olive]

First try to cook up a Holder to control the Lq norm by the L2 norm to a power and the L(2n/n-2) norm to a power

[sullen ember]

How do I use Hölders on a non-L1 norm? And how do I get 2 and 2d/d-2 as exponents? Sorry, I'm not really used to working with Lp inequalities yet and this was supposed to be an introductory exercise

[sonic olive]

Since q is between 2 and 2d/d-2 then there exists theta such q = 2theta + 2d/d-2(1 - theta). Then apply your usual holder inequality

[sullen ember]

ohh I see, thank you

[buoyant pike]

[ocean ether]

ok so

i've been tasked to solve the heat equation on an elipsoid with $u_t-\Delta u=0$ on $U\times(0,\infty)$, $u=0$ on $\partial U\times[0,\infty)$, and $u=c$ on $U\times{t=0}$

[untold delta]

cat bread

[ocean ether]

where $U={(x,y,z):\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq1}$

[untold delta]

cat bread

[buoyant pike]

<=1 or =1

[ocean ether]

leq 1

[buoyant pike]

If it's <=1 I would say inside an ellipsoid

[ocean ether]

U is an elipsoid

[buoyant pike]

Ok

[ocean ether]

obviously this is an easy separation of variables problem and $u=c_1e^{-\lambda t}w(x)$

[untold delta]

cat bread

[ocean ether]

and we use known solution of hemholtz on $\mathbb{R}^3$ where $w(x)=\frac{e^{\pm i\sqrt\lambda|x|}}{4\pi|x|}$ to obtain eigenfunctions

[untold delta]

cat bread

[ocean ether]

where would i go after this

[buoyant pike]

Ok let's consider the simpler case of the ball

I would not start with helmholtz

I would instead by finding an appropriate basis for r<=1 that satisfies the boundary conditions

Then solve the pde in that basis

[ocean ether]

wait why would i not start with hemholtz when this is a separation of variables problem

because i just need to find a basis that satisfies $-\Delta w=\lambda w$ in $U$ and $w=0$ on $\partial U$

[untold delta]

cat bread

[buoyant pike]

Well, how do you turn the free space solution into a solution with a boundary?

[ocean ether]

oh

ohh....

i'd try pulling a solution solely based on the radius because of symmetry

but i'm not really sure how that'd transfer over to an elipsoid?

[buoyant pike]

Well, try doing the spherical case with a non-uniform initial condition

[ocean ether]

wait a minute

this looks like it just

boils down to heat equation on a rod because of radial symmetry

[buoyant pike]

[ocean ether]

lmao i was thinking way too hard

i think this also works to the elipsoid too then?

[ocean ether]

yeah ok this was pretty trivial in retrospect because it boiled down to cookbook ODEs but i'm not sure how carry it to the ellipsoid

[buoyant pike]

Can you do a change of coordinates so that you have a sphere

And then do the appropriate change in the pde

[alpine carbon]

Hi, can anyone explain why this highlighted part is true?

[alpine carbon]

Hmmmm how do I prove this?

Source: https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/07%3A_Green's_Functions/7.02%3A_Boundary_Value_Greens_Functions

[alpine carbon]

I know there's that hint below, but how do I "evaluate the derivatives"?

[median forum]

The integral of the delta function is the Heaviside step function function which has a jump discontinuity of size 1

[alpine carbon]

Hi! How do I get the integral of the delta function here though?

[ocean ether]

ok so i'm actually not sure how we can bring boundary conditions into the mix with change of variables

[buoyant pike]

If it were a neumann boundary condition then it would be complicated

But it's a constant dirichlet boundary condition

The change of variables turns the ellipsoid into the balls and preserves the boundary so the boundary condition stays the same

[ocean ether]

so i have the change of variables $x_1=ax,y_1=by,z_1=cz$ on the ellipsoid $U={(x,y,z):\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq1}$ and this obivously reduces it to the ball and our heat equation turns into $u_t+\frac{\partial^2 u}{\partial x_1^2}a+\frac{\partial^2 u}{\partial y_1^2}b+\frac{\partial^2 u}{\partial z_1^2}c=0$

[buoyant pike]

Are you sure about those scaling constants

[ocean ether]

oops should be xa

[untold delta]

cat bread

[buoyant pike]

Anyways, once you find the correct PDE in transformed coordinates in the ball, then you consider an appropriate set of basis functions for the ball and proceed from there

[ocean ether]

damn so i cant appeal to the radial solution with the laplacian anymore

[buoyant pike]

Indeed not

[ocean ether]

fuck

wait so i can't appeal to the heat equation on a ball for insight?

[buoyant pike]

[shy narwhal]

ange's is terrifying

[sonic olive]

Does anyone here know about unique continuation/ nodal sets for parabolic pde?

[nova vault]

@sonic olive what is the op?

[sonic olive]

There is a parabolic version of the almgren frequency formula for global solutions, I would like to know if it is possible to get some localized version where I’m integrating over possibly a parabolic cylinder or heat ball, not the entirety of R^n. I haven’t really seen it in the literature and my attempts haven’t been working

[loud patrol]

is learning multivariable real analysis (like proofs based) better for the study of pdes

im distraught between taking multivariable real analysis or undergrad pdes

only single variable real analysis and lin alg and multivar calc for engineers and physicsist is required for undergrad pdes

[tall jolt]

what's multivariable real analysis? Like proof based multivariable calculus?

[loud patrol]

i guess

[loud patrol]

this is our multivariabel real analysis class

[tall jolt]

does it have a continuation course?

You should learn Stokes somewhere

[loud patrol]

no

idt

[tall jolt]

then you should self-study it

[loud patrol]

PDE class: A basic knowledge of the Riemann integral, including the
improper Riemann integral. Some exposure to the Lebesgue integral will be ideal, but it is not
strictly necessary as I plan to give a “soft” introduction to its essentials. Familiarity with the basics
of partial differential equations (PDEs) is helpful, but not mandatory. A good working knowledge
of advanced calculus and of the notion of partial derivative will be very helpful. Exposure to
integration by parts, aka the Divergence Theorem, will not be essential, as I will provide a crash
introduction, with lots of examples.

[tall jolt]

can't you take both courses? Learning multivariable calculus is essential for PDEs, so if you had to pick one I'd go for multivariable and learn it very well (also learn Stokes and all that)

I feel like the two courses would complement each other nicely

[loud patrol]

well just for next semester

cuz they overlap

[loud patrol]

this is actually grad pdes

[tall jolt]

idk you only gave the prerequisites

[loud patrol]

the multivariable real analysis and the undergrad higher level pdes

they're at the same time and day lol

[tall jolt]

mmh I mean you don't need to go to class I guess. If few students enroll maybe you can ask the prof to change the day/time of lectures

[loud patrol]

yea the math department is being dumb next semester

cuz why are they both 9 am classes and overlapping

[loud patrol]

i might just sit on the grad pde lectures tbh

not officially enroll just sit

ehhh i can enroll in the grad pdes

i have another problem im also doing physics and the applied pdes and physics math methods are at the same time as multivar real analysis, and undergrad pdes

i kinda wanna get out of the applied pdes

[tall jolt]

maybe you can try contacting the professor now, maybe other students have the same worries as you do. It seems reasonable to take both courses at the same time. but idk

[loud patrol]

department doesnt allow that

already spoke to them

maybe i just sit on grad pde lectures if prof is ok without actually enrollin g

[tall jolt]

oh I see. Yeah that's fully idiotic

[tall jolt]

mmh let me change my answer. If you really like PDEs, then yeah it seems alright to just go for the PDE course, but keep in mind that you'll have to learn multivariable calculus anyway if you don't already know it

[loud patrol]

the grad and undergrad pde professors publish together

so ik

idk

[loud patrol]

isnt stokes theorem also taught in differential geometry

[tall jolt]

Stokes theorem is really a multivariable calculus fact. Going from open subsets of R^n to manifolds (with some adjectives) is a straightforward patching process

[loud patrol]

hmm

i dont remember much of my engineering and physics multivariable calculus im tbh

we applied many of these theorems

[loud patrol]

it was about a year ago and i was going through some things that weren't conducive to my learning

[uneven helm]

Is anyone here an expert in asymptotic expansions?

[buoyant pike]

If you have a question, just ask

[broken hamlet]

I'm not an expert on PDEs/spectral theory, but it seems to be useful for what I'm doing, I'd like to check that I'm not doing anything insane.

Let $L_s$ denote a family of linear second order elliptic operators with nonnegative eigenvalues on a vector bundle $E$ over a compact manifold $X$. (by Weitzenbock, my operators are $-\Delta + D$ where $D$ is some first order linear differential operator). Suppose $L_s$ depends real-analytically on $s$. Let $m$ denote the multiplicity of the 0-eigenvalue of $L_0$. Then I think analytic perturbation theory gives me that locally, I have $\lambda_1(s), \dots, \lambda_m(s)$, which are eigenvalues of $L_s$, and these depend real-analytically on $s$. What I would like is to show that the smallest positive eigenvalue depends real-analytically on $s$, which I think I can show using a spectral gap argument. Is there a more direct approach to any/all of this?

[untold delta]

shingtaklam1324

[harsh veldt]

Is there anyone that can point me to a reference for an extension theorem from $H^{1/2}(\partial\Omega)$ to $H^{1}(\Omega)$? Thank you.

[untold delta]

emphatic_wax

[rotund jetty]

Proof of surjectivity of the trace operator is in Taylor's PDE book 1. He gives the argument for the half space, which is rather direct using Fourier transforms. The claim for domains should follow in some way by using coordinate charts

[harsh veldt]

i will look into this thank you

[copper finch]

Im stumbling over this in evans pde page 284.

I feel like I am misunderstanding things, since I don't get where the u is on the RHS?

[quick pagoda]

What’s (2)

[quick pagoda]

Well, for a given f, you know there is some u_f where <f, v> = (u_f, v)

And u_f is in H^1_0

So what about applying <f, u_f> = (u_f, u_f) = \int |Du|^2 + |u|^2

[copper finch]

yes. The LHS makes sense to me

but why is there no u in the right side. Shouldn't u be in place of the g? Why are there two g's

[sonic olive]

Cauchy Schwartz and then reabsorb into the left hand side

[quick pagoda]

Yeah I think that might work too, since it’ll look like L^2 inner product between the g and f functions

Maybe

[copper finch]

well i am still puzzled

[sonic olive]

You can also AM-GM

Write down what you get then apply an inequality to the rhs to match terms with the left hand side

[quick pagoda]

Ok so, suppose we bundle the f^i, g^i together as F, G for a sec

<u, u> = <f, u> = <F, G> \leq (<F,F><G, G>)^1/2 or wtv right?

If by <,> for these lists of functions means like pairing up corresponding indices

Unless I’m messing up Cauchy Schwartz because I’m an idiot

But <F, F> = <u, u> in H^1

So divide by that <F, F>^1/2 and you get like

<u,u>^1/2 \leq <G, G>^1/2

Square

This gets the desired inequality below (4), in your book’s numbering

I think

If it doesn’t work or isn’t clear then idk yell at me

[copper finch]

I think I understood it. I also wrote some stuff down and better got the relation between u and f here. Thanks.

[quick pagoda]

Nice nice

[pulsar basalt]

dull

[shy narwhal]

oof

[pulsar basalt]

i live in #numerical-analysis now

at least i think i do..

[buoyant pike]

[alpine carbon]

Hi! I don’t understand how to do this problem. I know you have to assume that G = G0 + H, where G0 is the free space Green’s function, I just don’t understand how to get H.

In general, I have no idea what the process for finding planar Green’s Functions. If anyone could recommend a guide/resource/textbook, that would be greatly appreciated!

[quaint herald]

There is no reason to expect a technique that works for an arbitrary domain. But in simple situations such as the ball and half-space, a powerful idea is to exploit symmetry. The free space greens function F almost works as a greens function for half space, apart from the boundary condition. So you need to add a harmonic function H whose boundary data (Neumann in this case, but it is the same idea if Dirichlet) cancels that of F. Consider the composition of F with a reflection about the boundary. What happens when you hit it with the Laplacian? What is its boundary Dirichlet and Neumann data (compared with that of F)? Now conclude by choosing H appropriately.

(If your domain was a ball, the symmetry is instead inversion through the boundary sphere, rather than reflection through the boundary hyperplane).

[alpine carbon]

Hi! Thanks so much for your reply! And sorry for my late reply. I saw it earlier but I didn't know how to respond, so I kept working on it.

Yes, I see now that if you reflect the Green's function over the boundary, it disappears when you take the Laplacian. Is this always the case for half planes? I could see it happening with y = x, because reflecting it gives an argument (x - eta)^2 + (y - xi)^2, and doing the calculations make it seem like it cancels out.

Just to clarify, what do you mean by "inversion through the boundary sphere"?

[quaint herald]

No problem.

Yes, the Laplacian is translation/reflection/rotation invariant, so translating/reflecting/rotating a free-space harmonic function will produce another one. Same for Greens functions, the singularity will just get translated/reflected/rotated.

By inversion, I mean the map on R^n \ {0} that maps x to x/|x|^2. It is like "reflection through the sphere |x|=1". You can show by a simple calculation that harmonic functions remain harmonic after you act on them by inversion, so you can use the same idea to solve boundary value problems on the ball in R^n.

[alpine carbon]

Got it, I'll try attempting a question with a boundary condition on a sphere. Thanks again for your help!

[loud patrol]

should i take undergrad or grad pdes

undergrad pde is at 9am and grad pde is 12pm ... the grad pde professor seems chill

idk if undergrad pde will be more computational but it requires introductory real analysis but the grad pde will definitely more advanced

[median forum]

Do you have more detailed course descriptions? What is your background?

[loud patrol]

yes

im not sure of the specific of the undergrad class

but i did email the grad prof

[loud patrol]

im taking a course on "advanced linerar algebra" which we're using linear algebra a second course by helene shapiro

and introductory real anlysis using understanding analysis by stephen abbott

here is the grad class description

undergrad class- prereqs are computation lin alg and diff eq and introductory real analysis

i spoke to the grad prof and he said he can give me resources for the prereqs if i decide to enroll

im like confused cuz i was gonna take multivariable real analysis but the professor for multivariable real analysis has not responded

[median forum]

If you've taken any multivariable I would definitely take the graduate one, looks very solid and approachable

Like not even multivariable analysis - just calc 3 type of stuff

[loud patrol]

i see i have done that in the past i just need a big refresher

which i can probably do over my winter break

would be doing grad and undergrad pdes at the same time be too much probably

im ngl i rlly dont wanna be studying pdes at 9am lol

[median forum]

Yeah I sympathize. Somehow the two only courses I want to take next semester are at 9 MW so I have to decide between them

[loud patrol]

well

im dropping classical and math methods 2 for next sem so i should supposedly have more time

[loud patrol]

for physics

[loud patrol]

well he said i may need to know the proof of the divergence theorem and the definition of partial derivative

is lebesgue integration hard to learn

[median forum]

These are not bad

[median forum]

IMO some aspects yeah. The way that it is used in the basics of pde stuff is fairly intuitive I think though

[loud patrol]

i think they talk about lebesgue integration in the multivariable real analysis but that prof has not responded and idk if i should email again

i think i would take the multivar real anlysis or the grad pdes

cuz ik multivar real anlysis is gonna be also very difficuly

the grad pde professor was actually answering my emails

[median forum]

That’s typically a good sign, though it doesn’t hurt to email the other professor again

[loud patrol]

would it be too much to be enrolled in both lowkey

i need to manage helath issues

im dropping math methods 2 and classical to take in the fall

well a grad class is worth two undergrad classes so prolly a bad idea

multivar real analysis

[median forum]

I think if you have seens 1 dimensional rigorous calculus and multi it is fairly simple to extend your knowledge to the multivariable setting

[loud patrol]

so i may not need to take am multivar analysis class rn

so

ive kinda heard more positive things with pde prof

is it better to get a letter of rec wtih a prof for a grad class or undergrad idk

as an undergrad who may apply to grad school in fall 2028

well it's more about what they can say about my abilities i guess if i do well

im also take the grad profs analysis and geometry class

[median forum]

The prof from the PDEs class has already demonstrated to be communicative and willing to word with you, so if you engage heavily with their class that will probably make for a good letter

[loud patrol]

yea i met with him in person to discuss the content

his research is in geometric analysis

[loud patrol]

yea im scared fro the multivar real analysis ill have trouble waking up on time well not that much but idk i have to walk 20 min

[median forum]

It's the same prof for both courses?

[loud patrol]

yes for pde

[loud patrol]

pde and geometric analysis r same prof

[median forum]

I would take both

[loud patrol]

they’ll prolly compliment each other i guess

hmm i can take the real analysis sequence like

next fall i guess

hopefully not at 9am

[verbal nebula]

Here's an interesting paper that just came out on the arxiv

https://arxiv.org/pdf/2511.09556

"Instantaneous Type 1 Blow-up and Non-uniqueness of Smooth Solutions of the Navier-Stokes Equations"

[buoyant pike]

https://arxiv.org/abs/2511.09556

[verbal nebula]

Yeah that's it!

We were just discussing this at my Fluids Seminar when I was supposed to be speaking. Instead of me speaking on a paper, my advisor and a postdoc were getting stressed just by the title, the abstract, and theorem 1.1

[stray fox]

What are their takes on this?

[verbal nebula]

It seems very strange. They didn't quite understand what the statement is, or why it works. It seemed to not agree with their understanding for 2D. In particular, the solutions exist for all time in 2D, but they have a blow up at any time T_* < T. It seems very strange indeed

[buoyant pike]

How odd

I agree that I don't quite understand the statement

And why it doesn't conflict with the previously established 2d existence results

[stray fox]

Hmmm.

Yeah I thought navier stokes is globally wellposed in 2D for smooth initial data.

[buoyant pike]

Ummmm

Hmmmm

Something about non-uniqueness of weak solutions?

No but they are saying that the classical solutions blow up

[verbal nebula]

Yes, this is the crux of the issue

[verbal nebula]

If the initial data $u_0 \in L^2$ and the vorticity $\omega_0 \in L^1 \cap L^{\infty}$ then the weak solutions are unique. This is Yudovich's result. But when you try to weaken this from $L^1 \cap L^p$ then things get messy. Chapter 8 of Majda and Bertozzi go through this, I think

[untold delta]

MoonBears-C-

[verbal nebula]

I think non-uniqueness is for Euler in 2D; whereas Navier-Stokes Equations in 2D will be unique. I've been trying to sort this out for a bit now

[fierce forum]

I haven't fully read it yet. But in Euler there's a certain richness to the weakened solutions over the strong ones at least from what I absorb from the work of Greg Eyink.

I won't be happy with any results contradicting this section of Majda and Bertozzi

[verbal nebula]

Yeah this paper by Cheskidov, Dai, and Palasek seems very strange indeed

[buoyant pike]

Indeed

[loud patrol]

the multivariable real analysis professor let me into his class

uh

idk if I should properly enroll into the grad PDEs then cuz it’ll probably be a lot of work

should i just ask the professor i can sit in his grad class

the term is about to end

[quick pagoda]

Why are time-varying domains so stinky

[last wind]

Hey guys! Can you help me solving this problem? I got it in university at calculus of variation course and have no idea how to solve it. The task is to find extremums of J[u]. It looks like Sturm–Liouville problem, but it lacks condition on u(1) and isoperimetric constraint acting only on the interval [0, 1]. Do you know any tips for solving this?

[buoyant pike]

Can you latex this

[buoyant pike]

Ok what have you tried

[proper knot]

use lagrange multipliers in banach spaces to absorb the L^2 constraint into the functional. Then use Euler-lagrange

[last wind]

Thanks!

[broken hamlet]

In Hamilton's book on Harmonic maps on manifolds with boundary, he ends up with an inequality of the form
$$\norm{f}{L^p{k+1/2}} \le C(1 + \norm{f}_{L^q_k})^{q/p}$$
for $q \gg p$, by working with polynomial differential operators and things like that

[untold delta]

shingtaklam1324

[broken hamlet]

Is this some general result?

Here, $f$ is a function on a compact smooth manifold $M$

[untold delta]

shingtaklam1324

[broken hamlet]

(and $L^p_k$ seems to be the geometer's notation for $W^{k, p}$)

[untold delta]

shingtaklam1324

[broken hamlet]

More generally, I would be happy with any result which says that a $L^q_k$ bound for all $q$ implies a $L^p_{k+1/2}$ bound

[untold delta]

shingtaklam1324

[verbal nebula]

So I met one of the authors, apparently their notion of weak solution is "very weak"

Weak enough such that the well known theorems do not apply

[buoyant pike]

They still have blow up for classical solutions in finite time though

[verbal nebula]

You know, this is what I think too. But when I'm at the table as a fucking 3rd year PhD student, I am not speaking up against authors, their collaborators, and experts in the field

I just smile and wave

[buoyant pike]

Speaking of, we have a PDE seminar today

Double DE Seminar: Nonuniqueness of Leray--Hopf solutions to the unforced incompressible 3D Navier--Stokes Equation

[loud patrol]

are PDEs at the graduate level more based on analysis than basic ode computation - how much ode should i know? i just took a computational engineering ode class with some mentioning of theorems

im taking real analysis rn

and also linear algebra

also multivariable calc?

what’s a good book for multivar calc that i can manage in a month

the grad pde prof just sent me some resources on analysis not necessarily ode or multivar calc like Riemann integral and lebesgue measure and integration

[median forum]

I think it is enough to mostly have seen existence, uniqueness, gronwall that type of stuff. PDE is mostly based around analysis because in general you can never actually get an explicit PDE so you have to do a lot of work to show that a solution exists in a suitable sense. This requires a lot of heavy analysis to deal with function spaces because the "simple" function spaces are in some sense not the amenable to using analysis or not natural for the PDE.

[median forum]

If you want to know the formal proofs maybe something like Spivak calculus on manifolds, but tbh I don't have good references. For quick stuff I think the appendixes from Evans PDEs (the earlier stuff not the functional analysis) are pretty good at getting you up to speed - in that you can use it as a list of things to know and try to learn most of the stuff (some of it is not necessary)

[loud patrol]

hmm

[median forum]

This is the exercise one of the profs I worked with when I started learning pde told me to do to make sure I knew "enough" about ODEs

[loud patrol]

did u see this?

yea the professor also has a chapter on L^p spaces

[median forum]

Yes, this type of analysis is the most fundamental in some sense and is very necessary for a good understanding of PDE. Understanding L^p spaces will be fundamental

[loud patrol]

he said i should know divergence theorem

i need brushing up on these

i took multivariable calc in a tough time in my life

[median forum]

Yeah the statement and the corollaries in Evans appendix C (integration by parts stuff) will likely be enough (when I first learned this my prof joked to us "most professional mathematicians don't know the proof of this"). I think spending more time with the measure theory stuff will ultimately be more rewarding. The multi stuff we be bashed into you during the class by the computations you need to do

[buoyant pike]

You've asked some variation of this same question multiple times now

[rotund jetty]

What reference shows the asymptotics of the spectral function $e(x, x, \lambda) = \sum_{\lambda_j \leq \lambda}|\phi_j(x)|^2$ for the Laplace Beltrami operator on compact manifold with Dirichlet or Neumann boundary conditions? If easier, what reference shows $e(x, x, \lambda) = O(\lambda^{d/2})$? I found this from a book, but I want to verify it with some other reference.

[untold delta]

L

[quaint herald]

Asymptotics for this are called the pointwise Weyl law. A seminal paper is Hormander's "The spectral function of an elliptic operator", although there were some earlier works by others. (And there are subsequent results sharpening the remainder estimate under various dynamical hypotheses on the geodesic flow of the manifold.)

If you merely want a big-O bound and not even the leading order asymptotic, there may be a cheaper method than usual arguments involving wave/heat traces. I haven't seen one myself though.

[rotund jetty]

But Hormander's paper assumes no boundary. The formula should be different if there is a boundary right?

Like for Dirichlet conditions, the function has to be 0 on the boundary

[quaint herald]

I think the asymptotic is of the same form (same leading term involving volume) but the proof is harder. I don't know the references off the top of my head, but will have a quick look. I know Ivrii has some of the sharpest results in the boundary setting, but I am not certain they are established for the pointwise spectral function, so I will doublecheck.

Check out the references mentioned at the bottom of p18 onwards in the text "Mathematics past and present: Fourier Integral Operators" (an awesome book btw, I have a hardcover copy).
I don't really feel like going through the refs right now, but they are the main developments in the proof of the Weyl law on compact manifolds with boundary, and I'd imagine that at least some of them (if not all of them) proceed via a pointwise Weyl law.

This stuff gets very technical if you want the asymptotic and sharp estimates on the remainder.

[rotund jetty]

is it easy to prove just the $O(\lambda^{d/2})$ bound uniform in $x$?

[untold delta]

L

[rotund jetty]

Agmon proves that $e(x, x, \lambda) = O(\lambda^{d/2})$ uniformly in $x$, but he assumes a bounded domain in Euclidean space. I'm under the impression that this implies it is true for general compact manifold with boundary, even though I'm not sure how to prove this.

[untold delta]

L

[quaint herald]

Well I don't know an automatic way of porting the result from domains to that on manifolds with boundary, but perhaps the proof can be ported with minimal changes. Otherwise you will have to look at other references.

[deft vessel]

I did PDE 1 without ODE and it was ight, but i did like PDE theory not a class on solving them yk?

[loud patrol]

i think it’s more pde theory tbh

[deft vessel]

If it’s more theory is just a completely different world

[deft vessel]

Functional analysis is like the basis of that theory like for the basics at least

Maybe ODE knowledge is more used on Fourier transforms but it’s still pretty easy ODEs

[verbal nebula]

Ok, it's not a Leray-Hopf weak solution. They only get instantaneous blow up in finite time by a non-local fourier mode/energy cascade argument, through a discretization

And their blow up is only from the right-hand side because the non-local interactions on the non-linear terms are from higher frequency things interacting with lower frequency things; however it's done in such a way that lower frequency things can't be decomposed to interact with higher frequency things

So there's an asymmetry in their method giving rise to the "Type I Instantaneous blow up"

I will go read the paper when I have time, but that was my conversation with one of the authors

[buoyant pike]

What does it mean to only blow up from the right

What does that mean for a hluid

[mortal slate]

Are there different ways to obtain H^{-1} inequalities?

[mortal slate]

I want this but for Lp

[buoyant pike]

We all want many things

Have you looked at the poincare inequality

[deft vessel]

I think there are some Sobolev einbettungen idk how that’s called

But look for Sobolev inequalities

[mortal slate]

Thanks!

[astral vine]

It has two different names: Necas negative norm theorem, or (Generalized) Wirtinger-Poincare inequality

[mortal slate]

This c, is it well known?

[buoyant pike]

[mortal slate]

I would like to find c independent of dimension, but this may be a lot

[buoyant pike]

The c in the previous inequality?

Or another c

[mortal slate]

Oh no, in Necas negative norm theorem (the first paper I see) it has some constant c

I dont know where to find a good source on Necas negative norm

[buoyant pike]

What other papers have you read about it

[mortal slate]

Nothing, I don't come from PDE's

But I appreciate the reference - has helped a lot

[mortal slate]

Do you have a recommended source on this in a book? Or is it mostly just from literature

[astral vine]

Look for "Necas Lemma" also. But it cannot be dimension-independent. Just because it involves the diameter of the unit ball at least.

[mortal slate]

I'm interested in the very specific case where the measure is isotropic

dimension free poincare inequalities have been a big study as of late and I wanted to experiment with this

[astral vine]

I truly don't know then

[mortal slate]

Yeah no worries

Was just looking for that kickstart! Cause we are snagging a lot of stuff from this field

[astral vine]

One way to prove it is to use bogovskii operators

So looking for the continuity constant of Bogovskii operators on thos isotropic Sobolev spaces (sounds like nightmarish, since Harmonic Analysis for differnet-measure Sobolev spaces sucks a lot)

[mortal slate]

lol damn

yeah

https://arxiv.org/pdf/2507.15495

This is the paper at hand that uses dimension free H^{-1} inequalities

[astral vine]

L^2 is always a bit apart it terms of estimates and accountances on other quantities for continuity constants.

[mortal slate]

The whole scheme a lot of the time is to introduce a random process on your measure, which converges into a Gaussian with a bit of work. Then analyze this process of integrals and use the "niceness" of the Gaussian.

In case you were curious

[astral vine]

Always good to improve my superficial knowledge

[mortal slate]

haha

for real

[astral vine]

@quick pagoda if you check on Arxiv. It is finally available.

[quick pagoda]

oooooh

I'll see if I can find it

located, thanks for notifying, I wont be able to catch errors but very cool

[astral vine]

If you have questions on the content feel free to ask. :)

[quick pagoda]

Thank you, though I doubt I'll have anything interesting to ask given my passing skill

[raven narwhal]

On a compact Riemannian manifold M of dimension at least 2 whose geodesic flow is ergodic (for example, if M has strictly negative curvature) prove Quantum Ergodicity: show that along a subsequence of Laplace eigenfunctions of density one the associated semiclassical/Wigner measures converge to the Liouville measure on the unit cotangent bundle S*M

[raven narwhal]

I saw this question in a paper but I'm not sure if it's correct

[quaint herald]

Wdym by "I'm not sure if it's correct"? This is the quantum ergodicity theorem, and is famous and has had many papers written about it in various contexts.

A pretty standard proof is in Zworski's semiclassical analysis text for example.

[raven narwhal]

Yeah I know the quantum ergodicity theorem is the standard result I just wasn’t sure if the phrasing was correct because It originally was actually the unique ergodicity version which is still a conjecture in general negative curvature I meant to ask for the QE statement not full QUE

[quaint herald]

The "density one" weakening of the conclusion, and the weaker assumption of "ergodicity" rather than strictly negative curvatures makes it the QE theorem, rather than the QUE conjecture.

[raven narwhal]

Oh right

The original wording accidentally stated the full QUE conclusion With the density one subsequence and just ergodicity of the flow it becomes the standard QE theorem thanks for the clarification

[quaint herald]

No worries.

[mortal slate]

So just curious, but is this what people would call a "regularity" estimate extending to L_p?

Just curious, PDEs isn't my main study.

[deft vessel]

anybody who can explain caccioppoli ineqs?

[astral vine]

This is something else but not completely unrelated.

[mortal slate]

what is the common term then for "extending" results to L_p?

[rotund jetty]

One way is by interpolation

[mortal slate]

but I only have an L^2 estimate, shouldn't I also need another estimate to interpolate?

[rotund jetty]

yeah, like an L^infinity or L^1 estimate.

[astral vine]

This does not work for any type of estimate.

[sweet epoch]

Do you have references/text about applications of weighted sobolev spaces to parabolic pde?

[astral vine]

See the work of Lorist, Veraar, Roodenburg, and Lindemulder.

And most importantly the references there in

like Krylov etc.

[sweet epoch]

thanks

[sweet epoch]

which krylov? I read chapters form his book on pde & sobolev spaces, but I want to understand better the weight approach. My problem here is that I am studying short time existence for quasilinear parabolic equations on compact riemannian manifolds. In the paper by Huisken and Polden on Geometric evolution equation, they use weighted parabolic sobolev spaces with exponential weight. My question is why they choose an exponential as weight, why not a power weight or something else?

[sweet epoch]

ok got krylov refs

thx

[grand trout]

Hi, i need some help with a wave equation problem

[buoyant pike]

Ok

Are you going to tell us the problem

[grand trout]

The boundary conditions u(x,0) and u_t(x,0) are intended to hold for all x in the real numbers

I know this can be done with alemberts formula but i want to solve it using forier transform

[buoyant pike]

Ok

What have you tried

[grand trout]

Still haven't done any paperwork yet, am gathering info

[buoyant pike]

Well

Try it first

And if you need help then ask

[grand trout]

Yea i have an idea

I just wanted to see if anyone has a maybe cool method to do it so

[grand trout]

Is this channel friendly?

[buoyant pike]

I hope so

[arctic whale]

So I know about the existance of Green function in the comapct with boundary case using the spectral decomposition of the laplacian i.e. if $v_j, \lambda_j $ is the spectral decomposition, then we $\Gamma(x,y) = \sum_j \frac{1}{\lambda_j}v_j(x)v_j(y)$ would be a green function satisfying the conditions 1,2,4. Does this construction also satisfy condition 3? Or is there a different construction that insures that 3 holds too?

[untold delta]

Co-aerA

[rotund jetty]

I'm pretty sure it doesn't satisfy 3 exactly, but it may satisfy it after multiplication by a constant. I know that 3 is how the Euclidean green function is defined, so condition 3 says that locally, the green function behaves like the Euclidean green function. This seems not easy to prove

[nova vault]

is this answered @arctic whale

?

cuz L is ^^ p much right

proving (3) is not easy it depends on how much kernel thry ya know

but technically 3 is correct

oh my bad l said (3) is not correct?

well, that not correct

(3) is true

[arctic whale]

wow is it really that hard to show? What I had in mind is to take a small geodesic chart around y so that the metric is an epsilion perturbation of the Euclidean metric and Christoffel symbols are uniformly bounded by epsilon.

Then maybe one hopes to show that the laplacian of such functions (Euclidean green functions) is not too far from delta_y

[nova vault]

sorry the hardness is not abiut intuitive prfs, sorry, i meant it's hard in rigor

like i have a prf in my head which uses some heat kernels on some reimman surf; but you hv the righ inuition fr sure

the elliptic regular stff is the annoying part

ya it's not hard my bad

sry maybe i am overcomplicating things

but 3 is correct

[arctic whale]

I mean yeah it's very believable of course up to some factor as L mentioned. I was just wondering if it follows from the other properties of the Green function on M, or if there is a construction of G on M that makes sure that 3 is satisfied.

[nova vault]

ya that's why i mentioned the elliptic regularity thingy

it's gonna bug ya

the "factors" ya mention are a spec case

but in the sense of the theorem as posted yes 3 is true

now i guess the new q is ^^

this will need some elliptic thry honestly

maybe there is someone smarter here

who can explain it simply

i can't

i am kinda bored and can write a shoet note to help

[dense vigil]

If u continuous on closed ball B1(0,0,0) and harmonic on open B1(0,0,0) where it is non negative ive to find the value u(1/√3,1/√3,1/√3) using that u(0,0,0)=0. Ive already shown that u=0 on B1/2(0,0,0) by harnacks ineq. Any hint?

[ocean ether]

how would i go about solving this

[buoyant pike]

What do you mean by solve

[ocean ether]

ok so we have a drop of fluid (D^+) on an ambient fluid (D^-) with shared boundary S that takes the form of a unit circle and they have different dielectric constants

i need to solve for the electric field in D^- (E^-) and D^+ (E^+) given these boundary conditions

[buoyant pike]

Solve meaning like

You have explicit boundary values?

[ocean ether]

yeah

[buoyant pike]

Ok you can solve a laplace equation right

[ocean ether]

of course

[ocean ether]

it's just that the condition is kinda throwing me off

[buoyant pike]

Which condition

The dirichlet or the neumann

[ocean ether]

actually let me think about it a little longer

[ocean ether]

yeah i still have no idea

[buoyant pike]

Ok what have you tried since yesterday

[ocean ether]

i tried applying a solution in polar coordinates but that still doesn't solve my issue with $\Phi^+$ and $\Phi^-$ both not being fixed functions and depending on each other

[untold delta]

.

[hallow pumice]

Holy hell, I remember when catbread was doing calc 2, now you're doing PDEs

[verbal nebula]

To be fair, freshman year to junior/senior year is only 3 years. Which really isn't that long

[hallow pumice]

3 years huh? Yeah it's not that long

[verbal nebula]

It is strange. It's like 1 year of freshman university math is worth 3 years of HS math. 1 year of upper division is worth two years of lower division. And one year of graduate school is worth something like 2-3 years of upper division

[pine rover]

So one year of grad school is worth 12-18 years of high school math. Checks out

Unironically I think that's probably about right

[buoyant pike]

Greetings and salutations

Consider the velocity field $\mathbf{u}=u_0\sin\theta\mathbf{e}{\phi}$ on the sphere where $\theta$ is the colatitude and $\mathbf{e}{\phi}$ is the unit vector in the longitudinal direction

[untold delta]

守沢 千秋

[buoyant pike]

I am trying to determine what the material derivative of this is

So $\frac{D\mathbf{u}}{Dt}=\pdv{\mathbf{u}}{t}+(\mathbf{u}\cdot\nabla)\mathbf{u}$

[untold delta]

守沢 千秋

[buoyant pike]

My collaborator has written [\frac{D\mathbf{u}}{Dt}=u_0\left(\frac{D\sin\theta}{Dt}\mathbf{e}{\phi}+\sin\theta\frac{D\mathbf{e}{\phi}}{Dt}\right)]

[untold delta]

守沢 千秋

[buoyant pike]

Just using the product rule

Next, the claim is that $\frac{D\mathbf{e}{\phi}}{Dt}$ should be nonzero because $\mathbf{e}{\phi}$ is the local unit vector in the longitudinal direction and this is changing as the fluid moves

[untold delta]

守沢 千秋

[buoyant pike]

Is this reasonable?

[waxen bobcat]

Changing as the fluid moves?

[buoyant pike]

u is the velocity of a fluid

On the sphere

The e phi unit vector on the sphere is different at different points on the sphere

[waxen bobcat]

Hmm, not sure about this argument, but I got (u • ∇) u = -(u_0^2 / 2) sin(2 θ) e_θ if this makes sense

(using the christawful symbols)

[buoyant pike]

My collaborator gets that u dot nabla u is u_0^2 tan(theta) e_theta

Curious

[waxen bobcat]

I think it should be intuitive that (u • ∇) u vanishes at the equator (θ = π / 2), whereas it's not defined for the expression your collaborator got, but I might be mistaken

[buoyant pike]

Oh wait

tan(theta) for theta latitude, not colatitude

In colatitude tan(theta) should be cot(theta)

[waxen bobcat]

Ah, okay

[buoyant pike]

Which of course still does not match up

[waxen bobcat]

Yeah, hmm

[buoyant pike]

😵‍💫

[waxen bobcat]

Well, in any case, it's nonzero

[waxen bobcat]

Does it make sense that it explodes at the poles, though?

[buoyant pike]

The poles are always bad

Shallow water equations on the sphere (which this is for) have tangent of latitude so the equations explode at the poles

[waxen bobcat]

I guess so if u = u_0 e_phi, but we have u = u_0 sin(theta) e_phi

[buoyant pike]

You can't even define the unit vectors at the poles

Anyways what I'm really trying to figure out is the correct form of the shallow water equations on the sphere

They can be written as \begin{align*}\frac{D\mathbf{u}}{Dt}&=-f\mathbf{e}_r\cross\mathbf{u}-g\nabla h\\frac{Dh}{Dt}&=-h\nabla\cdot\mathbf{u}\end{align*}

[untold delta]

守沢 千秋

[buoyant pike]

D/Dt is the material derivative

f is the coriolis parameter, 2 Omega sin latitude

g is gravity

u is fluid velocity

h is fluid surface height

Omega is 2pi/day

We can also write this in vorticity divergence form

So defining $\zeta=\mathbf{e}_r\cdot(\nabla\cross\mathbf{u})$ and $\sigma=\nabla\cdot\mathbf{u}$, we have that \begin{align*}\pdv{\zeta}{t}&=-\nabla\cdot((\zeta+f)\mathbf{u})\ \pdv{\sigma}{t}&=\mathbf{e}_r\cdot\nabla\cross((\zeta+f)\mathbf{u})-\Delta\left(gh+\frac12\mathbf{u}\cdot\mathbf{u}\right)\end{align*}

[untold delta]

守沢 千秋

[buoyant pike]

However, my collaborator did a derivation and found that the equation for vorticity is the same

But for the divergence equation, they have an additional $-\mathbf{u}\cdot\mathbf{u}$ tacked on at the end

[untold delta]

守沢 千秋

[waxen bobcat]

I have to digest this a bit

[buoyant pike]

To test this, we can consider a simple solution

Namely u=u_0 cosine latitude

Now, you also need to specify a height field

One paper presents gh=\Omega u_0+u_0^2/2

My collaborator gets gh=(Omega u_0+u_0^2/2)cos^2 latitude

[buoyant pike]

This matches the equation set without the additional -u dot u

[waxen bobcat]

I see

[buoyant pike]

And this matches the equation set with the additional -u dot u

But like

They can't both be correct right

[waxen bobcat]

Yeah, so, you're verifying which one is equivalent to the original problem

[buoyant pike]

Yes, so we look at this original equation set

And try to determine what the material derivatives are

[waxen bobcat]

How does that help you exactly?

[waxen bobcat]

Can't you just compare how the paper and your collaborator got this vorticity divergence form?

[buoyant pike]

Well we can determine which height field is correct to make this is a steady state solution

[buoyant pike]

The paper doesn't present a derivation

[waxen bobcat]

Ah

[loud patrol]

how do i not get cooked in my grad pde class next semester???

ohhh the prof gave me notes on lebesgue integration

and some measure theory

[buoyant pike]

And to do this, we need to determine if including the derivatives of the unit vectors is the correct thing to do

[waxen bobcat]

Hm?

[buoyant pike]

Well, we need to determine Du/Dt right

[waxen bobcat]

But what do you mean exactly by including the derivatives of the unit vectors?

[buoyant pike]

Is De_phi/Dt 0 or not

[waxen bobcat]

Oh, for a steady state solution

[buoyant pike]

Yeah

[buoyant pike]

I think my own computation agrees with this

[tepid widget]

Suppose that $u$ is a harmonic function on the upper half-plane $\mathbb{R}\times[0,\infty)$ and that $u(x,0)=0$ for every $x\in\mathbb{R}$. Also, suppose that for every $x\in\mathbb{R}$, we have $\lim_{y\to\infty}u(x,y)=0$. Note in particular that we aren't assuming (a priori) that $u$ is bounded. Does it follow that $u$ is bounded (and therefore identically zero)?

[untold delta]

Gustav

[median forum]

I think so as you just do a Cayley transformation and then look at the disc

[median forum]

@tepid widget does this make sense?

[buoyant pike]

Ok so for the whole ordeal I was going through

I think I've found the problem

This vector calculus identity needs additional metric terms when used in spherical coordinates

[waxen bobcat]

so which one is correct? with or without the -u•u?

[buoyant pike]

They’re the same

As in

The minus u dot u is correct

[buoyant pike]

In particular

$\nabla\cdot((\mathbf{u}\cdot\nabla)\mathbf{u})=(\mathbf{u}\cdot\nabla)(\nabla\cdot\mathbf{u})+\nabla\mathbf{u}:\nabla\mathbf{u}+\mathbf{u}\cdot\mathbf{u}$

[untold delta]

守沢 千秋

[buoyant pike]

Do you want to try to christoffel symbols this

[ionic forge]

What's the diff between PDE and Adv PDE channel 🥺

[buoyant pike]

Feel free to read the channel descriptions

[turbid scaffold]

Can someone give me some motivation for what exactly a paraproduct is and why they’re relevant to pdes?

[raven narwhal]

Think of a paraproduct as a way to make sense of multiplying rough functions by separating low–high high–low and high–high frequency interactions via Littlewood Paley decomposition the key point is that in PDEs nonlinear terms often involve products that aren’t classically defined at the regularity you have and paraproducts isolate the parts that are well behaved

They’re essential in nonlinear estimates like Navier Stokes and quasilinear equations because they explain exactly where derivatives are gained or lost and turn ill defined products into controlled operators

[astral vine]

Very nice explanation

[quaint mural]

May I ask, is the #odes-and-pdes the correct channel to ask about theory of existance and uniqueness of solutions for ODEs using the theorem of prolongation of solutions, Picard and Peano or is it best here even if this is for PDEs?

[crystal wolf]

Probably better in the channel you linked :)

[celest dust]

Hi! I've been studying "Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems" of Hebert Amann. In page 20, he establishes a theorem where he characterizes being normally elliptic with being the minus generator of an analytic semigroup. There, he cites his book "Linear and Quasilinear Parabolic Problems: Volume I Abstract Linear Theory", which was not published when he wrote that. I'm not able to find this result in this book. Does anybody have a reference of where to find a proof of this result for higher order problems? Thank you very much!

[quaint mural]

Well I was just curious because the channel is linked as early university but this class uses a lot of real analysis (albeit not central) so I'm not entirely sure if people there will be able to help me, but thanks

[buoyant pike]

You might ask in #advanced-analysis if necessary

[turbid scaffold]

Thanks I saw something in Taos notes on paraproducts about this bony paraproduct decomposition. This seems like a significant property and relevant to what you’re saying. This is helpful.

[ornate peak]

I’ve never studied PDEs, but I have a background in operator algebras, probability theory and I have studied ODEs, but not recently and I have taken several courses on mathematical physics. Would it be reasonable to take a course on stochastic DEs, or would this be an uphill battle of extremes? This is at the grad level mind you.

[buoyant pike]

Are you familiar with measure theory

Like

realyl really familiar

[ornate peak]

Yes…. I’ve taken the course… I didn’t pass the comp but that’s not a huge deal… should’ve memorized the Hahn Decomposition theorem… I’ll try it again in my masters. But yes, now I work with Haar measure and the like so no I’m not worried about measure theory

But I see what you mean.., the way of thinking will be.., in that paradigm I see

[raven narwhal]

No worries mate!

[tired axle]

I am looking at an exercise in Reed-Simon volume 1. The question asks for a proof that the equation [-\Delta u +\abs{u}u+u=f] with $f\in C_c^\infty(\mathbb{R}^\nu) and $u\in L^2(\mathbb{R}^\nu)$ has a unique solution. The wording of the problem is such that it's clear that the want this result to follow from the strict convexity of the associated functional
[T(u) = \int \abs{\grad u}^2 + \frac{1}{3}\abs{u}^3 + \frac{1}{2}u^2]
However, I worry if these two formulations are completely equivalent. I would be convinced if we knew that any solution to the PDE must necessarily belong to $H^1(\mathbb{R}^\nu)$. However I don't see why this would be true. The quadratic nonlinearity in the equation is really obstructing any obvious elliptic regularity argument that I can see

[untold delta]

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

[tired axle]

it seems like all we can really say is that $(1-\Delta) u\in L^1$. So $(1+k^2)\hat{u}(k)\in C_0$. This doesn't seem that useful. I think you can get something in $\nu=2$ by Hausdorff-Young but this method has no chance of generalizing to arbitrary dimension! Is this even really true for arbitrary $\nu$?

[untold delta]

Buncho Spheres

[proper knot]

you can add to your functional the term -int u(x)f(x) and then show that it attains its infimum (and you also allow your functional to attain the value +infinity since a priori you don't know enough regularity)

you want to use weak lower semicontinuity (assuming that this got somewhere introduced before)

[vestal iron]

Nope Simon doesn't talk about l.s.c. that I thinks only brezis does

[proper knot]

I read through some online pdf of it, and here they talk about l.s.c.

but whatever version one uses, surely they talk a lot about weak convergence ...

[tired axle]

Yes so semicontinuity and coercivity is fine for me as far as showing that a solution exists. Strict convexity even implies that a unique H^1 \cap L^3 solution exists. But the PDE formulation is much weaker!

In particular I’m struggling to believe that every L^2 solution of the equation is a critical point of the functional

A priori T(u) may well be infinity, so being a critical point would not make sense without further regularity

[tired axle]

RS has an appendix on the calculus of variations where this is all discussed

[vestal iron]

So it has l.s.c and conjugate function as well?

[tired axle]

Just lower semicontinuity

[hallow pumice]

Prox spotted in math server

[sonic olive]

You can fix |u| + 1 to be your potential term and apply an apriori estimate for gradient u (for example using the pde for mollified u) the problem is that this works when |u| is in L^{n/2} which in this case is up to dimension 4 or if you have an estimate on the L3 norm of u.

You can also try to do a Calderon zygmund estimate and again you run into the problem where the right hand side is in L1 which is borderline.

Assuming that u is in L2 and L3 fixes all of your problems in both of these approaches, I feel like this is a valid assumption as you have mentioned, fatous lemma gives you this for minimizers anyway.

These are also artifacts of that fact that your leading order term is the laplacian and not some variable coefficient equation. For those it is possible to have distributional solutions that are not H1

[tired axle]

What's the arugment you're using that would work if you knew that u was in L^3? The argument I'm thinking of puts uhat in L^p for all p>d/2. Hausdorff-Young is relevant so long as d/2 <=2 or d<=4, as you say. This then puts u in L^p for all 2<= p<d/(d-2) . So |u|u is in L^p for 1<=p<d/(2d-4). So |u|u is in L^2 for d<12/5, so it seems like you can't even get it to be in L^2 in order to use the usual sobolev regularity unless d<=2. But presumably you're thinking of something different as there is no mollification involved here

being in L^3 puts |u|u in L^3/2 so I guess you can try hardy-littlewood-Sobolev? I never remember the exponents for that off the top of my head

but also finally while i'm very interested in an argument that brings you up to H^1 from just L^2 \cap L^3, i'm curious then if you have purely distributional solutions in the pure L^2 case

[tired axle]

hello hello!

[proper knot]

do you currently want to show that a solution exists, or that strong solutions are unique?

[tired axle]

i'm convinced that a solution exists. I'm even convinced of uniqueness if we assume a sufficient amount of regularity. But I would like to understand if uniqueness holds for solutions that are merely L^2

[sonic olive]

Assuming u is smooth and u is L2 and L3, integrate by parts

[\int |\nabla u|^2 \leq C\int |u|^3 + |u|^2 + |f|^2 ]

Now if u is just L2 and L3, then mollify u and you will get the same estimate for the mollified function. So by compactness, the u is in $H^1$.

[untold delta]

maark

[serene mica]

Im reading this paper and maybe i am just not seeing something, but I have no idea how Definition 1.1 makes sense given Theorem 1.2 (both on page 2). Definition 1.1 requires integrating u\theta against a test function with compact frequency support, but I dont think the paper proves u\theta \in L^1 or even in some Sobolev space, so this definition makes no sense. Help me out, is there something I am missing?

https://arxiv.org/pdf/2403.16800

Definition and Theorem in question

[tired axle]

Ahhh that makes sense!

[serene mica]

fwiw, I think this paper is kind of just a mess. I looked at the official published version (not arxiv version) and they show that the integral with the nonlinearity goes to 0 as you send q -> infinity....but that would force \theta = 0, which makes the solution trivial

I doubt she would give a shit, but thinking of emailing Mimi Dai about this...cause...something is fishy

[tired hollow]

Cfl condition ? Dirichlet Neumann and robin

cfl diffusion explicit u=2deltat/delta x~< 1/6, f=0
Initial condition, for example, when the temperature fills the cube or we have u0= sin pi x sin pi y sin pi z, the Dirichlet boundary conditions or all=0, u(t0 y z) u(t x 0 z) and u(t x y 0) homogeneous
Neumann or partial derivative of u=0
Robin or alpha u+ beta derivative of u= g(t) limit

I have a problem with the diffusion

Matlab dosent give me each u value

Idk if it's the code or the equation

[arctic whale]

How do we get log-concavity of the first Dirichlet eigenfunction on convex domains in Rn ?

[ornate peak]

It’s not the measure theory actually.. it’s pretty reasonable… it’s the assumed knowledge of physics and comfort with stochastics that’s scary actually

[turbid scaffold]

Can anybody recommend some stuff to read about the applications of jet bundles to pdes?

[quick pagoda]

There’s Krasil’shchik (Geometry of Jet Spaces and Nonlinear Differential Equations), though I think it’s a bit old and idk if it’s quite the direction you want. Hairer has stuff on using generalizations from his regularity stuff to study SDEs, which I’d think is the right direction but maybe a bit beyond what you want.

[turbid scaffold]

I feel like that is the direction I want because I came upon the subject by seeing a paper by vinogradov on something called secondary calculus. But I figured this was old like you said and was hoping for something more modern.

I wanna learn about the characterization of pdes as submanifolds of jet bundles

[quick pagoda]

I think it might be in there, maybe in Jost geometric analysis, maybe in Saunders’ jet bundle book

I know some geometric analysis things say jet bundle things about fibers, maybe Lewis’ geo analysis on analytic manifolds, but I don’t know that book well

Taylor has a lot in it where’s L

[deft vessel]

hi! I dont really understand how the determinant of that is supposed to be read

can someone help?

[buoyant pike]

Is v a vector

[deft vessel]

nope

[ornate peak]

Does anyone know of a good book to learn Brownian motion? Or just a thorough treatment of it, it is assumed knowledge going into a course about it… a tad surprising.

[median forum]

I like Partzch and Schilling. You might also try asking in the advanced probability channel

[median forum]

What is the curly H function space

[quick pagoda]

Does øksendal cover Brownian?

[ornate peak]

True… most of the assignment problems are solving PDEs… and measure theoretic results but that’s a good point

[ornate peak]

I ordered it… I hope it’s good

[mortal slate]

yes

[upbeat bolt]

I was recommended "Lectures on the H Principle" by a grad student in a seminar I was in

Actually it's "Introduction to the H Principle" by Eliashberg and Misachev

[turbid scaffold]

Woah this book looks stuffed with info thank you!

[short barn]

The proof my professor gave last semester of the GNS inequality was different from the standard one given in Evans, Taylor, Gilbarg-Trudinger, etc. but seemed conceptually more insightful. Unfortunately I didn’t take notes and I don’t know where I can find this proof. Any recommendations?

[sonic olive]

How did the proof go? Did it use the coarea formula and isoperimetric inequality? Fundamental theorem of calculus spam? Fundamental theorem of calculus only once plus harmonic analysis integral estimates?

[short barn]

Fundamental theorem of calculus a bunch of times, I think we proved it for some endpoint case and then used some interpolation without calling it interpolation

[sonic olive]

I have a feeling that it is the same proof that is in evans then but probably using better notation

[short barn]

Or maybe I don’t know what I’m talking about and it doesn’t hold for the endpoints idk I’m not very good with thi stuff

Haha you might be right

[sonic olive]

Yeah that proof is classically very hard to write down

[short barn]

Do you have any recommendations?

[sonic olive]

Tbh just try to prove it yourself, that one is actually very straight forward once you get passed the notation slop

Start in 2d

If you can do it in 2d and 3d then you will be able to do in n dimensions

[short barn]

Ok I’ll try

[sonic olive]

If you know gmt then I would recommend the coarea formula proof as it gives you sharp constants, but this calculus proof assumes nothing besides calculus and basic measure theory

[short barn]

GMT = geometric measure theory?

Ok I’ll check that out after trying to work this out

Thanks for your help

[sonic olive]

Np good luck

[short barn]

Ok I think I realized why my professors formulation looked so much cleaner, he just used induction instead of integrating that product of integrals n times. But I now realize it’s pretty obvious how to do it

[elder ice]

Im tryna learn until advanced pdes what do I need if im starting from scratch like someone with no knowledge of math

[buoyant pike]

Calculus, multivariable calculus, linear algebra, real analysis, topology, measure theory, functional analysis, fourier analysis

[mortal slate]

Lmfao

That made me chuckle

[somber mountain]

what mathematics is your major.

[somber mountain]

my major is functional analysis and fractal interpolation

[turbid scaffold]

I’m not familiar with this system of majors. Usually we would say mathematics is a major. Are you asking about someone’s specialization? Like I took my comprehensive exam in algebra.

I’ve never heard of someone’s major being fractal interpolation(or ever heard of that topic)

[somber mountain]

sorry

[dusk dove]

Don't troll here

[proven bough]

devid u got a undergraduate role wyd

[thick vault]

Hi , Ihavent take a PDE course , i would like to know whether there are standard techniques to solve these type problems

[median forum]

This is first order, so it is generally not a bad idea to write out in terms of the coordinate derivatives and then try to use characteristics which is often a good technique for first order equations

[cloud thistle]

seconding method of characteristics ^

[surreal epoch]

Hello, any references where I can read about the square root of the Dirichlet Laplacian? I would like some with explicit computations. Thank you very much!

[quick pagoda]

this has some stuff in it you might want, maybe?

[jaunty briar]

Hello everyone, I was going to try to publish a paper for pdes and I was wondering if anyone would have some suggestions on what to publish

[vestal rampart]

Something that solves a gap in the literature is usually what ppl publish

[upbeat bolt]

You should check out these equations on fluid dynamics I heard abt they’re called Navy stoked or something

I bet if you could prove long time existence and uniqueness of solutions that would be a paper that’s not too bad

[astral vine]

There are shit tons of ways to PDEs

[astral vine]

There are no explicit formulas for square root for the Dirichlet Laplacian excepts in very basic geometries : spheres, half-spaces, squares/hypercubes etc.

[quick pagoda]

PDE
Explicit
Pick one

[astral vine]

Then there is Ange always struggling with life and explicit representation of solutions with spherical harmonics

[quick pagoda]

There’s a reason he’s struggling

Even spheres and balls are struggles 🥀

[surreal epoch]

Even in intervals would be interesting for me as I am building my understading around the topic

[surreal epoch]

Thank you very much, I will check it.

[buoyant pike]

Sob

[astral vine]

olev

[buoyant pike]

omg recently I got back reviews for a paper

And one of the reviews was like

Why don't you solve the PDE when the initial condition is ill posed

sob

[waxen bobcat]

ol

[loud patrol]

how do you generalize that a one d solution to a linear pde like the transport question to multiple dimensions

[median forum]

Do you mean the transport equation in multiple dimensions?

[loud patrol]

yea basically

how do a generalize the solution

[median forum]

I think for the simple case it is just characteristics:

[median forum]

Suppose I have the equation $\partial_t v(t,x) + b(t,x) \cdot\nabla_x v(t,x)= 0.$ We seek solutions of the form $v(t,x) = f(x(t)).$

[untold delta]

zeke

[median forum]

If we say that $x'(t) = b(t,x(t))$ the gives an ODE for the curve for which we know solutions exist. The PDE gives $df/dt = \partial_t f+ \nabla_x f \cdot x'(t) = \partial_t v + b \cdot \nabla_x v = 0,$ so the value of $f$ is constant along each characteristic curve $x(t)$. For any point in the domain where you want a solution you (try to) find such a curve connecting it back to the initial domain

[untold delta]

zeke

[median forum]

In particular if $b(t,x) = b$ a constant vector, $x(t) = x_0 + bt,$ so then the solution is just $v(t,x) = g(x - bt)$ where g is your initial data

[untold delta]

zeke

[median forum]

Does this answer your question @loud patrol ?

[loud patrol]

Yes

[loud patrol]

yea we got that solution in class for the one dimensional case thank you so much

[loud patrol]

is that divergence the nabla sub x or just gradient with respect or x

im not familiar with notation that well pdes is totally new to me

[median forum]

grad wrt to x

[loud patrol]

these are basically the same question then lol

[loud patrol]

can someone explain shocks to me

like for example if the transport equation has a nonlinear coefficient x^2 instead some constant vector

is it like asymptotic or something idk

[median forum]

Shocks arise from when you do this and you have multiple curves that connect the same point to your initial data, and the value of your solution on those curves is different, so you get a discontinuity

[loud patrol]

i see

ok like

i was told to compute

$$\begin{equation}
\begin{cases}
u_t + x^2 u_x, & \textbf{x} \in \mathbb{R} \backslash { 0}, t\in \mathbb{R}\
u(x, 0)= \varphi(x), & x\in \mathbb{R}
\end{cases}
\end{equation}$$

[untold delta]

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

[loud patrol]

when i was trying to find the characteristic curve i got $\alpha(s)=(x(s),t(s))=(\frac{x}{1-xs}, s+t)$

[untold delta]

n

[loud patrol]

but i dont understand

so for the solution i got $$u(x,t)=\varphi(\frac{x}{1+xt})$$

[untold delta]

n

[loud patrol]

this has some weird asympotic behavior right?

[loud patrol]

thats wym by discontinuous