advanced-pdes

[jagged adder]

Has a shit ton of questions and has a solution book online too ig

[buoyant pike]

Strauss does not have decent theory

[jagged adder]

That’s why I suggested to supplement a lot of the theory with Evan’s once you’re more familiar with content

[buoyant pike]

To some extent every question in strauss is the same as well

Separation of variables/fourier series

[jagged adder]

Well I guess you have to supplement most of the theory with Evan’s lol. But I find Evan’s hard to read without having any background

[jagged adder]

I mean fair enough but it’s still practice?

[median forum]

Besides Evans I like some parts of rauch and Taylor volume 1. In some respects they are more gentle than evans and more linear in path

[rotund jetty]

Evans book, Folland's book, and M. Taylor's volume 1 are good for theory.

[frosty haven]

https://bookstore.ams.org/amstext-54/

[buoyant pike]

Did you use this book

The order of topics presented in the table of contents is a bit strange to me

[frosty haven]

i found it when i was looking for something deeper than strauss but not as deep as evans. it doesn't assume you've had measure theory or functional analysis, just some real analysis. i thought it was alright, but i didn't go through any other books thoroughly.

[heady silo]

anyone here started reading Evans recently? thinking of taking the plunge

[frank prism]

I think the first few chapters take a little getting used to (specifically step-to-step what he is doing in each estimate and the level of fluency he's expecting from the multivariable analysis), but overall it's a great book to go for

[carmine anvil]

Me, I've finished upto the end of the laplace/poisson eq section I think

[median forum]

I highkey recommend skipping most of 3 and 4

[verbal nebula]

I've read it off and on for a while now. Chapter 2 is a fantastic chapter

I'd really recommend going through the multivariable calculus identities in the appendix, especially the different forms of integration by parts

[pulsar forge]

i do think its better to pick more specialized books after chapter 5 of evans, but its still a handy reference

by the time you start learning elliptic equations for example, there is more classical and thorough books on the topic, same thing for parabolic/nonlinear topics

first 3 chapters of evans are fantastic tho

[heady silo]

thanks everyone

[exotic void]

Does anyone know why B^1_\infty, 1 is called critical for the incompressible Euler equation? I originally thought it meant that Besov norm is invariant under the scaling λu(λx, λ^2t) like we learn for the Navier-Stokes, but after doing the computation that doesn't seem to be the case.

[astral vine]

This is not the correct scaling this one is for Navier-Stokes

\dot{B}^{1}_{\infty,1} is like W^{1,\infty} somehow

try the scaling 1/λ*u(λx, t/λ)

or λu(x/λ, λt)

[untold delta]

Functionanatolysis

[astral vine]

(or homogeneous Besov spaces, tho)

[exotic void]

I see. What's the motivation behind this scaling? Unlike Navier-Stokes having only one scaling, Euler can accommodate a few different scaling.

Is this scaling related to transport equation? I have an inkling that people bulid solutions from that

[astral vine]

essentially

[exotic void]

That really helps! Thank you

[buoyant pike]

[uneven helm]

Are there any phd students/people who have a phd from the USA here?

[rotund jetty]

yes there are

[uneven helm]

When you guys applied, did you already have pde research under your belt? During my bachelors I never saw single research project (over the summer) that had non-numerical pdes

However, reading forums I get the sense that mist people seem to have some sort of publications at the point of applying

[astral vine]

Did something happen Ange ?

[buoyant pike]

Someone asked and then deleted a question

[tidal urchin]

The goal of this is to find how many training steps a tanh activated single layer FFN would require to converge. For simplicity I used a limit to effectively make the loss requirement 0 but in practice I would make some variable as a threshold. However, this makes the math a bit easier.

We seek $u(x,t)\colon \Omega\times[0,T]\to\Bbb R$ satisfying

$$
\frac{\partial u}{\partial t}(x,t)
;=;
D,\Delta u(x,t)
;-;
\eta;\bigl(,\tanh\bigl(u(x,t)\bigr);-;f(x)\bigr),
$$

for $(x,t)\in\Omega\times(0,T]$. Here:

Domain:

• $\Omega=(0,1)\subset\Bbb R$
• $t\in[0,T]\subset\Bbb R^+$ is “training time.”

Unknown

• $u(x,t)$ is the pre-activation field. The network’s predicted output at $(x,t)$ is

  $$
  y(x,t);=;\tanh\bigl(u(x,t)\bigr).
  $$

Parameters

• $D>0$ is a diffusion constant
• $\eta>0$ is the learning‐rate.
• $f(x)$ is a given “target” (our calibration dataset interpolated as a function on $\Omega$).

Initial condition

$$
u(x,0);=;u_0(x),
\qquad x\in\Omega,
$$

where $u_0(x)$ is some prescribed initialization (e.g.\ small random or zero).

Boundary conditions (homogeneous Dirichlet for concreteness)

$$
u(0,t);=;0,
\quad
u(1,t);=;0,
\qquad t\in[0,T].
$$

6. Ending conditio

   $$
   \lim_{t\to\infty}\tanh\bigl(u(x,t)\bigr)
   ;=;
   f(x),
   \qquad x\in\Omega,
   $$

   i.e.\ as training proceeds, the network output converges to the target.

I want to solve for a value t that satisfies the PDE

[untold delta]

Apolloiscool

[tidal urchin]

oh also the computation shouldn't just be training the model and counting the steps. Anybody could do that. I want to approximate this in a faster way than training a massive model

[median forum]

What is your question exactly?

[tidal urchin]

"How can I determine the number of training steps of using SGD for a single layer FFN that uses a tanh activation function given the learning rate, number of parameters, and the dataset?"

I think it's easier to define the convergence of the model on the dataset to be a vector field where where the vectors represent the gradients of the function in back-propagation. So that's why I posed it in #advanced-pdes because I want to solve for t which shows the number of steps the model needs to move to converge the loss to 0

[median forum]

I don’t know anything about ML but I’m confused what solving for t means in the context of your equation - do you mean solve for u?

[ornate flare]

Could be misinterpreting, but I think the goal is to find a T>0 such that $|\tanh(u(x,T))-f(x)|<\varpesilon$ for some suitable tolerance $\varepsilon$. Is that accurate @tidal urchin ?

[untold delta]

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

[tidal urchin]

Exactly

The issue is that there is not really a closed form solution as far as I know

because it's stochastic

[ornate flare]

Without knowing the literature I can’t really comment, but I would just try to discretize the PDE and simply plot the error over time. Another thing you could try (although probably with little success) is linearize ur pde around the equilibrium solution $u(x)=\tanh^{-1}(f(x))$ and consider the linearized equation for the error $v(x,t)=u(x,t)-u(x)$. Maybe u could find some upper bound this way, but again I’m not sure.

[untold delta]

peter_legare

[tidal urchin]

you can't plot the error over time because that's simply just running the massive model which defeats the purpose

[ornate flare]

sorry lol I was only thinking about the pde not the context

[tidal urchin]

No worries

It's truly a tricky problem

[novel kayak]

This might be obvious but I somehow cant establish the Gronwall-type inequality here. The Gronwall inequality in Lemma A.2 is the integral form. I assume that to use it, I have to bound |grad X| by
1 + int_0^t |grad u|_Linfty |grad X| ds
somehow, but im not able to arrive at this.

I've tried working over matrices all the way or expanding everything, but I keep getting an extra factor of n popping up from |I| = n

The text is Bedrossian and Vicol's "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes Equations", this is quite literally the third page

[astral vine]

Actually integrating your differential equation you obtain

X(t)=X_0+ integral from 0 to t of Du(s)^T • DX(s)ds

X_0 being the identity

Choose an algebra norm on matrices

So the norm of Identity is 1

Then apply Triangle inequality, norm algebra property and Gronwall

You are done:)

[novel kayak]

But are we allowed to pick the norm here
I was thinking |grad X| is the Frobenius norm

[lilac barn]

#advanced-pdes message Follow the discussion here

[pulsar forge]

congrats on honourable role

[lilac barn]

Oh wow!! I didn't know I got the honourable role!

No more incentive to help people now 😎

[royal quail]

I love cars

[novel kayak]

Oh wow its been discussed before, thanks! I'll look at it when I'm done with classes

[astral vine]

deserved, buddy

[neat spoke]

Any intuition as to why the barrier condition is a natural thing to ask for a domain to satisfy (as in Perron's method for the Dirichlet problem)?

[copper finch]

If I have two W^1,2 functions on R and I want to take the integral of u'v in the weak sense, then by integration by parts how exactly does the value at infinity vanish so that we get the integral of u v'

[pulsar forge]

approximate by functions that vanish at infinity then take limits

[lilac barn]

Follow the discussion here: #real-complex-analysis message

[rotund jetty]

How I would do it: The bilinear map $B(u, v) = \int u' v , dx$ from $W^{1,2} \times W^{1, 2} \to \mathbb{C}$ is continuous. Form another appropriate bilinear map and show it agrees with B on the dense set $C_c^{\infty} \times C_c^{\infty}$.

[untold delta]

L

[copper finch]

So my lecture notes are saying "polar coordinates" here, but how is there polar coordinates in 3d? Isn't it spherical then? I also don't quite get the b^2 / epsilon^2 part. Where does that epsilon^2 come from in the second line

[jagged adder]

For the second question: you have that |x|^2 < epsilon^2

So you just flip the inequality and pull it out

[copper finch]

why does it get bigger then? Shouldn't it be |x|^2 > epsilon^2

[jagged adder]

No cuz x is in the ball of radius epsilon

[carmine anvil]

yes, and thats what you have since x is outside B(0,eps)

[jagged adder]

oh yeah, I can't read

[copper finch]

Thanks!

[grizzled hornet]

hey, jfyi this is an advanced channel which deals with partial differential equations. If you need help with a basic math question, please use a help channel.

[noble gate]

Is there actually an application of the dirichlet problem and brownian motion connection, or is it just cool

I guess you can do monte carlo on the expectation

[quaint herald]

Maybe I misunderstand your question, but it lets you bring PDE tools to bear on Brownian motion problems.
One example I've seen is the study of the expected time that a particle travelling in BM first enters some subset of a manifold (e.g. a small geodesic ball about a point of radius r). Asymptotics of this expected hitting time as r-> 0 are directly related to asymptotics for the Greens function.

[quick pagoda]

Also lets you twist some geometric data out of it, motivation-wise

So things like spectrum of the solution map corresponding to connectedness data

Not like you couldn’t do fourier-y things anyway, but it can motivate things

[shy narwhal]

useful

[pliant cove]

I'm having trouble coming up with the initial and boundary conditions for a time-dependent wave equation on an infinite cylinder. It's been a while since I've worked on PDEs. Does anyone have any advice?

Basically, I'm trying to work on a harder problem, but wanted to start simpler to work my way up

If someone does respond to this, please ping me because I might miss the notification otherwise.

[low glen]

Hi, would any of you be familiar with the study of stationary inhomogeneous solutions of the Vlasov equation by any chance? I would like to know which papers constructing them and studying their asymptotic stability are relevant. Thanks in advance.

[buoyant pike]

What is the Neumann to Dirichlet operator for the Laplace equation in a ball

[dim kayak]

@wise estuary Like bro

Some mfs wayyyy too smart

I'm not one of those

[dim kayak]

The Neumann-to-Dirichlet (NtD) operator for the Laplace equation in a ball is a boundary operator that maps given Neumann boundary data (normal derivative of the potential) to the corresponding Dirichlet boundary data (the potential itself) for harmonic functions inside the ball.

[rustic oak]

Please do not use AI to answer people's questions

[dim kayak]

mb og

[verbal nebula]

I don't really have an answer, so what I did was google it. I found this (which you could also find, by a simple search)

https://arxiv.org/pdf/1202.3605

[buoyant pike]

Well I mean I know what it is

I want some explicit description of how to compute it

[verbal nebula]

All I see are theorems being proved, rather than explicit formulations

[median forum]

Can you not differentiate the explicit formula from Poisson kernel to get the result? So it will be some integral operator that is a derivative of the poisson kernel

After you take a limit mb

[buoyant pike]

Well that gives dirichlet to neumann

But how can you go the other way

[median forum]

Oh my bad

[buoyant pike]

Lol no worries

[broken hamlet]

Try the book by Paternain-Salo-Uhlmann

[buoyant pike]

Geometric inverse problems?

[quick pagoda]

Oh yeah, I need to figure out norm bounds for these Neumann-to-Dirichlet-like things

[buoyant pike]

Sharp

Do you know how to write these down

[quick pagoda]

I’ll be yoinking that

[quick pagoda]

No, I’m actually currently attempting to do exactly this lmfao

[buoyant pike]

I tried looking through it but nothing stood out to me as relevant

[quick pagoda]

For unrelated reasons

[buoyant pike]

Well most of the discussion is for domains in R^2

[quick pagoda]

I kinda need balls in R^n

[buoyant pike]

I only need a ball in R^3

[quick pagoda]

Well, do you know the R^2 one

I’ll be trying to write down some things, but uhhh don’t have high hopes for me, I only really want norm bounds

[buoyant pike]

Well in 2d I know the Green's function for the laplace equation for neumann boundary conditions so I don't need a neumann to dirichlet operator

[quick pagoda]

Well, mind passing that green function along since I don’t know that one

But uhhh, I’ll let you know if I stumble across anything

Kinda unlikely I find any such expression but yk

[buoyant pike]

Oh I found a source online with the green's function for the neumann poisson

Unclear if this is correct though

Their 2d does not match this 2d one: https://math.stackexchange.com/a/2273858

[quick pagoda]

That’s uhhhhh sussy

[buoyant pike]

I recently read a paper where they have the green's function for the biharmonic equation on the sphere wrong

They got it wrong because they tried to match another paper which had it correct but with a different definition of the dilogarithm

People will publish anything

I guess in principle the 2d one shouldn't be too hard to test

Maybe I'll do that tonight

[broken hamlet]

Ye

[buoyant pike]

Do you have any recommendations for which sections I should look at

[broken hamlet]

not really, just thought it might be relevant

[buoyant pike]

[quick pagoda]

Pde slop mines….

[rotund jetty]

Could you use the laplace eigenfunction expansion to get some abstract formula for it, then try to look up a closed form?

[quick pagoda]

If we know u is harmonic on some bounded open set O and cts up to the boundary w/ smooth boundary data, then is there some epsilon > 0 where u is harmonic on an epsilon fattening of O?

[minor mulch]

what if boundary conditions are 0 and you extend it to 0 outside of \overline O

like say O is an open ball

nvm 0 boundary doesnt work

but maybe you can do something like this to get a counterexample

like extend it by making it constant on rays

[median forum]

What if you take O_\epsilon to be the fattening and then take U to be O_\epsilon - O

Ah you then get mixed boundary condition hmm

[minor mulch]

i feel like this cant possibly be true

[quick pagoda]

Well I’m trying to figure out if I can extend it harmonic-ally, so I don’t think the ray thing would do it idk

[median forum]

Ah this is the same as an overdetermined problem on the complement of O

[minor mulch]

oh so ur asking if there exists a harmonic extension?

[quick pagoda]

Yes, if there’s some epsilon depending on u, where u is able to be extended harmonic on O_\eps

[minor mulch]

cant u just solve laplace eqn in O_eps setminus O

[median forum]

I think this shouldn’t be true because you are essentially just prescribing both Neumann and dirichlet data

[median forum]

No you need normal derivatives to agreed as well

[quick pagoda]

Well, even if you can’t for one eps, how do you show you couldn’t for any, for starters

Secondly, you’d be harmonic in O and O_\eps - O separately

But I think you might run into issues possibly with weird averages on the boundary?

[minor mulch]

nvm im being dumb

[median forum]

Yeah should just be this

[quick pagoda]

Maybe it is just that easy, but like, consider meromorphic functions on C

We love weird singularities that can’t be avoided, so not every epsilon words

Otherwise like, every harmonic function would be entire so uhh

So, that’s like something that could pose an obstruction, but on the other hand, I can’t think of something offhand that has smooth boundary data and all that but can’t be extended

(And O is bounded)

[median forum]

Maybe just a power series that converges everywhere on the boundary of its disc of convergence?

[quick pagoda]

Right, but it also needs to be nicely smooth radially too

Or uhh not radial maybe but on the boundary but wtv

[median forum]

Yeah then I guess I’m forgetting what the right notion of smooth on the boundary is

[quick pagoda]

That’s a reasonable point, a bit less meaningful if it’s not a nice smooth boundary ig hmmm

I suppose there shouldn’t be much hope of some dense collection of harmonic functions that extend beyond a Lipschitz domain or wtv

[minor mulch]

can you just do a nonanalytic boundary condition

any harmonic extension would be analytic on the boundary of O

ig you meant analytic bc

[quick pagoda]

That’s actually very reasonable as an obstruction

Just gotta throw something like that and you’re cooked

But yeah I’m just kinda hoping there’s some dense choice of boundary data that lets things extend

This is probably more cope than hope though

[buoyant pike]

Ok so testing these

$f(r,\theta)=r^4\cos(4\theta)$ is a harmonic function

Ok

[untold delta]

Angetenar

[buoyant pike]

The normal derivative on the boundary is $\partial _r f(r,\theta)=4r^3\cos(4\theta)$

Right?

[untold delta]

Angetenar

[buoyant pike]

The Green's function from MSE is $G(x,y)=\frac{1}{2\pi}\ln\abs{x-y}+\frac{1}{2\pi}\ln\abs{x-y/\abs{y}^2}$

[untold delta]

Angetenar

[buoyant pike]

Where x is on the boundary and y is a point in the interior

So in principle, $r^4\cos(4\theta)=\int_{S^1}G(\theta_1,y)(4\cos(4\theta_1))d\theta_1$

[untold delta]

Angetenar

[buoyant pike]

Where y=(r,theta)

Ummmm

$\abs{x-y}=\sqrt{2-2r\cos(\theta_1-\theta)}$

[untold delta]

Angetenar

[buoyant pike]

$\abs{x-y/\abs{y}^2}=\sqrt{2/r\cos(\theta_1-\theta)}$

[untold delta]

Angetenar

[buoyant pike]

How am I supposed to do this integral

Let's see what mathematica says

[buoyant pike]

Ok

This is what mathematica gives

This looks not correct

[buoyant pike]

Ok I have verified that this is not correct by plugging in r=1/2

Unless I did something wrong

Sharp you should check

[tall jolt]

where can I read about a qualitative analysis of the schrodinger operator associated with the hydrogen atom in higher dimensions?

for example: is there a formula for the energies (aka eigenvalues)? If not, are there asymptotics? Can we calculate the dimensions of the eigenspaces?

wait nvm I think I can obtain the answers

[tall jolt]

We are interested in the Hamiltonian -Delta-q/r^(d-2) in R^d, where q is some constant. Let us assume that eigenfunctions H psi=E psi can be written as psi=R(r)Y(theta), where r is the distance to the origin and theta are angular coordinates. Delta can be rewritten

Delta=partial^2_r + (d-1) r^(-1) · partial_r + r^(-2) · Delta_S

where Delta_S is the Laplacian on the d-1 sphere. Y should be an eigenfunction of Delta_S, the eigenfunctions of Delta_S are known (eg. https://en.wikipedia.org/wiki/Spherical_harmonics#Higher_dimensions) and the eigenvalues are of the form -l(l+d-2) with l=0,1,2,... Let Y=Y_l be such an eigenfunction.

Then the equation reduces to the equation in the screenshot

also assume d>=3, d=2 and d=1 seem degenerate

when d=3, the standard case, people assume R is analytic. They write R=e^(-sqrt(-E))L(r) (it should be the case that E<0 for the non-continuous part of the spectrum or something), and assume that R (i.e., L) is analytic (I have no idea how to prove that this is actually the case atm).

The differential equation provides a recursive formula for the coefficients of L, and one can check that R is in L^2 if and only if L is a polynomial. If a_k is the largest nonzero coefficient of L, then the vanishing of a_(k+1) (which should imply the vanishing of the rest of coefficients, maybe) gives relations between l, k and E, so it's possible to give a formula for the possible E's and the dimension of the eigenspaces.

But when d>4 solutions are never analytic (if there were, we would have q=0, something we don't want, of course). For d=4 there are some particular analytic solutions, but only for very specific values of q (but they aren't in L^2 I think)

so it seems that higher dimensions are way subtler 💀, unless I'm missing something

[tall jolt]

maybe for d>4 it's not even (essentially?) self-adjoint, so it's unreasonable to expect a nice theory?

oh okay nvm there are no negative eigenvalues

strange

[tired hollow]

What is the Schrödinger operator?

[tired hollow]

Did you mean the Hamiltonian operator?

Its eigenvalues are positive since they represent energy Ievels.

[frank prism]

this agrees with heuristics I'm vaguely aware of from quantum chemistry, there's a massive industry of making appropriate chemical approximations for many electron Schrodinger equations (and equations like the one you derived) to recover asymptotics

this is more or less what every (atomic-level) computational chemistry program is doing, if there were easy asymptotics available without needing to do this, many chemists would be out of work

e.g. this is more or less the basic thinking that leads to considering things like Born-Oppenheimer approximations (which often let you recover asymptotics at certain energetic critical points, which are the states chemists usually care about), which are somewhat chemically justified but I'm not aware of anything rigorous on it (but far from my domain, I'd bet there is some stuff, I only talk to chemists here and they don't tend to care about what PDE people do too much)

If you would like to know anything specific on this end, my twin sister is a physical chemist who does this stuff, I can ask for pointers to literature

[verbal nebula]

Discussing this paper https://arxiv.org/pdf/2508.19590

I was pointed in this direction https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

@astral vine I'd love to discuss this a bit further to see what your input is. Basically one of my mentors is saying that Tao's argument that these scaling invariant arguments are unlikely to work

[astral vine]

I know some of Myong-Hwan Ri's work

Still trying to get what's actually happening in the preprint

BUT

This is the more reliable attempt for years

Like it is not just because of invariant scale

[verbal nebula]

Alrighty, I'll try to look at it a bit deeper

[astral vine]

The point is to prove that there exists for any data a uniform minimal time of existence

could it be the Planck time constant

the scaling allows to go from uniform time opf existence to global solutions

you still need to prove this uniform time of existence, and the idea seems to look at the frequency interaction at the tetra exponential (or log^4 depending on the point of view) level

But even if in the end it is not true, the idea is a bit new, and to add credit to it, Myong-Hwan Ri already produced known solid math work.

(might be fully wrong in the end, tho)

Being just scale invariant with respect to Standard Sobolev-Besov norms does not work, but here there is a bit more than that

[verbal nebula]

?

[buoyant pike]

?

[astral vine]

?

[low glen]

?

[harsh veldt]

Hello. DOes anybody know what the trace lifting theorem is? Any reference on the statement and proof for this?

[astral vine]

Yes I know it, I have an explicit reference for the extremely general statement which is unreadable. A proof for result in the case of smooth domains can be found in Giovanni Leoni's book.

Which version are you looking for

I cna provide references for other cases

[harsh veldt]

The one involving H1 and H^{1/2}

I have not found it in books so far lol

[astral vine]

Okay for this one I don't know a specific reference, but I can providean easy proof

"easy" I mean

Just basic Fourier analysis

here is an easy proof : https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03132-2/S0002-9939-96-03132-2.pdf

actually you need interpolation theory for the sharpness of this one but sicne you are only interested in H^{1}-H^{1/2} you cna remove this argument

[harsh veldt]

Okay I will read through the paper. Thank you!
I needed a function v in H1 for any w in H^{1/2} for the proof I am working on...

On the boundary...

[astral vine]

this is true for Lipschitz domain

[harsh veldt]

okay. That works... thank you!!! I think I am already assuming that anyway

[astral vine]

(you can even go below Lipschitz using measure geomtric assumptions)

[harsh veldt]

I also needed it for some regularity results.

thank you for being helpful!

[astral vine]

The very abstract and fully general statement can be found in the book by Jonsson and Wallin 1984.

[harsh veldt]

what is the title?

pretty sure my mentor will ask. lol

[astral vine]

But I really cannot recommend it

"Function spaces on subsets of R^n"

p.109 and p.197

[harsh veldt]

I will look into it. I'm pretty new in the field so it might really be unreadable for me

[astral vine]

It is also almost unreadble for specialists

be careful that for high regularity fro the trace H^{s,p}, $s>1+1/p$ the trace theorem does not hold in the usual sense

You need more regularity on the boundary to have the trace theorem in the expected "usual way".

[harsh veldt]

Thank you! At least for now I am only working on s = 1. I will keep this in my notes though

[astral vine]

H^{s}, s>3/2 still has a trace in H^{s-1/2} but if the boudnary is say lipschitz then H^{s-1/2} of the boundary does not match with the usual definition (and this is necessary for the statemetn to hold)

[harsh veldt]

So does it require a smoother boundary?

[quick pagoda]

Yum

[astral vine]

If you have say a C^{1,a} boundary the trace theorem with ontoness H^{s} to H^{s-1/2} holds in the usual sense whenever 1/2<s<1/2+1+a

C1/Lipschitz you cna take a=0.

This is not that much well known and some bullshit has been written down in the literature

[astral vine]

In which university are you again ?

Missouri Minnesota ?

[quick pagoda]

None of the above

[astral vine]

Okay because some of the people there are very well aware of these issues

very strong people

[quick pagoda]

I believe it

Hoffman save me

[astral vine]

hahahaha

[quick pagoda]

I really need to learn those details better, though unfortunately they currently elude me

[astral vine]

before grinding those heavy details people should learn first how to do PDEs on smooth bounded domains the correct way. (Not talking about you particularly) But yet there are still subtleties about just function spaces over R^n, people are writing shit about.

Like homogeneous Sobolev norms and inhomogeneous Sobolev norms always equivalent for compactly supported distributions. This is utterly wrong.

This is implicitly assumed in so many papers on elliptic PDEs it will give people headache

Just on H^{-1}(\R^2) one the most important space, the result if false here.

[quick pagoda]

Yeah I believe it

Some garbage I’m cooking up is kinda problematic for specifically this reason even

[astral vine]

what are you cooking exactly ?

[quick pagoda]

Some horrible model theory x PDEs things

So a lot of really terrible function spaces on bounded domains

And, currently, trying to exactly figure out some of these “norms on the trace” type things for smooth domains

[quick pagoda]

(This is probably somewhere extant in the literature though)

[astral vine]

I see now why you are interested in my yupcoming preprint

[quick pagoda]

Indeed

I’m not so far gone as to be looking at too much weird weighted function spaces, but a lot of terrible L^1_loc multipliers

[astral vine]

In 4 weeks I will start a full lecture on Besov spaces and trace theory.

[quick pagoda]

Oooh

Way above my pay grade but fun

[astral vine]

I also do multipleir theory

[quick pagoda]

I think one of my stinky norm problems is a Hardy space-ish thing

Which is awful because I don’t understand a thing about those

[astral vine]

I did read a bunch of papers and a 800 pages book about for this upcoming preprint

[quick pagoda]

Yeah there’s a lot of horrible (semi?)norms flying around based on bad multipliers and oscillating

But that can probably be erased if I was less stupid

[astral vine]

I will try to put the typed notes up to date and to share it in some places

Probably put it on Arxiv if I am happy enough of the final result.

[quick pagoda]

I’d definitely grab and stash them, much less likely I will understand them though

[astral vine]

This will be about notes with extrinsic motivations

this is designed for master students, the goal is for them to learn about the topic, not to kill them

[quick pagoda]

What a legend

Actual exposition

Could not be Guy David

[astral vine]

Guy David, is too much of a model and an expert

[quick pagoda]

Guy David vs Guy David

[astral vine]

Title: Besov spaces.

Description: When it comes to the precise study of boundary value problems and (parabolic) evolution problems in PDEs, it turns out that the appropriate framework to describe the nature of the objects is actually the setting of Besov spaces. This lecture will provide their construction from many various contexts, mostly motivated by the study of (non-)linear PDEs beyond the Hilbert space setting and will contain:

• Introduction of Besov spaces, inhomogeneous and homogeneous on the whole space, the half-space and the link with standard Sobolev spaces;
• A short introduction to interpolation theory
• Trace theory and Besov spaces as the exact trace space on domains with a boundary;
• Besov spaces as the optimal space for initial data in linear evolution equations.

[quick pagoda]

Seems like an explanatory blurb to me

[frank tide]

Suppose I have a bilinear form on V=H^1(0,l) from a weak formulation B(u,v) and I define the operator
(Au,v)=B(u,v). If I can write
B=B^1+k^2B^2 and associate A^1 to B^1, A^2 to B^2 where A^1 has bounded inverse and A^2 is compact, then Fredholm theory tells us that A is invertible for all k outside of some discrete set.

Now say I have some conforming finite element space V_h, I can go through a very similar argument and show that for fixed h, A_h is invertible for all k outside of some discrete set. But how would I show that as h->0, the k I say “initially chose” doesn’t fall into a discrete set where A is no longer invertible

I was thinking like if you could write
A_h=A^1_h+k^2A^2_h, then define K_h=(A_h^1)^{-1}A_h^2, you ‘d have
A_h=A_h^1(I + k^2 K_h) and so A_h is invertible if and only if I+k^2 K_h is invertible, which means you need -1/k^2 to not be in spectrum(K_h) for all h. That is, spectrum(K_h) has to be similar to spectrum(K) as h->0. Are there any spectral approximation theorems you could use to show something like this or is there a direct theorem I can reference?

Also sorry if this isn’t the right channel, I wasn’t sure to choose between this, analysis, and numerical

[tall jolt]

Is anyone familiar with the proof of Nash embedding (for compact manifolds) from Taylor 3?

Here the Laplacian is the Laplacian on R^k/Z^k with the standard coordinates (so the Laplacian on R^k acting on Z^k-periodic functions)

Why is F non-linear? Why is DF(0) invertible? Why do we require r to not be an integer?

[lilac barn]

F is nonlinear because Q is bilinear and F(w) ~ Q(w,w). So F(w+ v) = Q(w,w) + Q(w,v) + Q(v,w) + Q(v,v) which is not F(w) + F(v)

DF(0) being invertible is the invertibility of laplacian -1 on this domain

[tall jolt]

yeah so I'm asking, why is laplacian-1 invertible in that context?

[astral vine]

This is related to Sobolev and Besov spaces theory

1 - Laplacian (or eq. Laplacian - 1 ) is invertible on B^{s}_{\infty,\infty}(TT^k)

B^{s}_{\infty,\infty}(TT^k) = C^{s}(TT^k) when s not an integer

the equality fails for itnegers

[lilac barn]

Evans PDE 5.6.3 Thm 6 combined with 6.3.2 Thm 5

Moreover, if you look in 6.2.2 Thm 3 example, he shows that you can fix mu ≥ 0. So set mu = 1 and since L = - Lap then it corresponds to your case.

[tall jolt]

will check Evans in a bit

but doesn't Evans work exclusively with open subsets of R^n with Dirichlet boundary condition? How is that applicable to the situation of the torus?

for the same reason, I don't understand why 6.3 Theorem 5 is relevant

[lilac barn]

I think this should not be an issue as you can transform into other boundary conditions as he states.
However, in your case, you can also just use Fourier transform to write the solution as Inverse fourier transform of ( fhat/(1 +k^2 + n^2)) and then you can get the spaces mentioned using the usual embeddings

This immediately makes use of the Torus domain as the -laplacian is k^2 + n^2 where k,n is the Fourier vairables

[tall jolt]

where does he state that, and what boundary conditions?

[lilac barn]

6.1.2 Other boundary conditions

[tall jolt]

but the boundary conditions they give don't seem relevant to the case of the torus?

[lilac barn]

I suppose yes, but I don't think there has to be made much changes to redo the entire theory for the torus case (I might be wrong). In any case, you can use the Fourier transform method I mentioned above

[tall jolt]

that seems like an extremelly inefficient way to learn a subject, but ok

btw can you spell out what you mean by the Fourier transform method? I'm not very familiar with that. Isn't the Fourier transform only defined in Z^k in the first place?

(or give a reference)

[lilac barn]

You can show that −Δ has the multiplier |ξ,η|²=ξ²+η², in other words −Δf=g is just f̂=ĝ/|ξ,η|². Now this is not invertible as it is singular at 0. But (−Δ+1)f=g implies f̂= ĝ/(1+|ξ,η|²) is not singular at 0 anymore so you can invert this and carry out whatever estimates.

If you are not familiar with Fourier transform, then a reference would be Elliptic PDE book of Trudinger. That should have the result you are interested

[tall jolt]

why doesn't Taylor say anything about Besov spaces, or does he? Do you have a reference for this?

[astral vine]

There are some minor subtleties

[astral vine]

This approach as such only works for Lp spaces 1<p<\infty

But using these formula you cna infer a formula that works for all p, and prove boundedness

[astral vine]

Bescause no explicit references as far as I know, except maybe the two first book on Function space Theory by Triebel maybe

(This however not how I would prove it myself)

[tall jolt]

Have you read taylor? Is it supposed to be self-contained, or is the reader supposed to fill in the gaps? So far, I have seen no proof of Nash's embedding theorem, only an outline. But it's true that I haven't read what's before in Taylor's books

[tall jolt]

prove what? How would you prove it?

[astral vine]

invertibility of Laplacian - 1 on C^{r} with values in C^{r+2}

[astral vine]

I did read a wide part of Taylor, don't remember of this

and you should be precise about what you call nash embeddings

[tall jolt]

smooth isometric embedding into some R^n of a compact Riemannian manifold

[tall jolt]

What's your prefered proof then?

[hidden coral]

can you just do this with fourier transform

[astral vine]

Not on L infty type spaces as such.

[untold delta]

Andrew

[quick pagoda]

Well, what about on the edge of that ball where it’s constant?

[untold delta]

Andrew

Andrew

[quick pagoda]

Well, is that still a local maximum

Well, consider how it’s flat at the top, since you know it’s constant around a local max, so when would it fail to be a local max in a larger ball containing it

If that point on the boundary is still where it’s a local max, then you can extend where it’s constant

Don’t elliptic things have minimum principles too

Well, draw a small ball around a local minimum, then apply principles from elliptic things on that ball

I think there’s min and max things there right?

So if it’s a local min or local max you should be fine

So, what could stop it from being a local extremum on the boundary

Yeah, but in that case, it might just go from max to a min locally

[sonic olive]

There is unique continuation for elliptic pde with lipschitz coefficients

[quick pagoda]

That’s the only possible obstruction

[sonic olive]

Garafola Lin- unique continuation and Ap weights

Something like that

I can send you a link in private messages I’m not sure if the discord will auto ban me or something for posting a link in chat

Yeah makes sense, show that the solution is constant on a neighborhood of the local max/min then use unique continuation to prove it’s constant on omega

You can write it of the form of a linear equation though

Because you already have a solution, like you said the coefficients would be C1

Yeah this is why studying linear pde is important even for nonlinear people

Where is this exercise from btw?

[tall jolt]

what is known about navier-stokes in 2d? Is the analog of the milennium problem solved? What about navier-stokes on surfaces? Is there a nice text where to learn this

[buoyant pike]

Solved in 2d yes

The incompressible 2d case is covered nicely in Vorticity and Incompressible Flow by Majda and Bertozzi

Not sure what class of surfaces you have in mind, but locally 2d Riemannian manifolds are well posed as well I think

[tall jolt]

any smooth surface

[buoyant pike]

Well R^3 is a smooth surface isn't it

[tall jolt]

mmh wtdym by "locally"?

[tall jolt]

nope? I mean 2d manifolds

or yeah, with a Riemannian metric

[buoyant pike]

Oh also 2d navier stokes potentially with compressibility was solved by ladyzhenskaya I think

[tall jolt]

thanks for the reference. Btw, this is more on the physics side, but if some solutions do blow up, what should we blame it to? One obvious deficiency of these models is that fluids are made up of particles and aren't actually continuous. But there's also relativistic effects which at high energies maybe should be taken into account. Could relativistic effects tame fluids enough so that they don't blow up?

I have no idea about this, maybe taking relativity into account makes this a million times harder lol

[buoyant pike]

The fluid assumptions will break down long before the state of the fluid reaches a relativistic state

Large velocity -> high temperature -> gas -> continuum assumption breaks down

[tall jolt]

what if we consider relativistic fluids anyway, are fluids expected not to blow up then? I'm not exactly sure how you would formulate the problem or if it even makes sense tho

[buoyant pike]

I'm not quite sure what it means for a relativistic fluid to not blow up, computational relativistic MHD is used to model black holes which have blown up by any reasonable definition

[tall jolt]

but it's the space time that's blowing up

like you could possibly have blow up of fluid without spacetime blow up

maybe, idk what I'm talking about really

[quick pagoda]

Ask a physicist

Isn’t there a book like Bedrossian-Vicol?

I dunno what’s in it but I think it has fluid things

[buoyant pike]

Bedrossian Vicol is incompressible only

[tall jolt]

I'm a bit familiar with the Cauchy problem in general relativity, idk if it's related

[broken hamlet]

Also like continuity methods lol

[copper finch]

Is there good literature on pde solving with operator/spectral theory? My lecture notes have examples here and there, but I'd like a whole book on it

[astral vine]

depending on which kind of PDEs the books could be quite different

Like Parabolic/Elliptic PDEs from fluids have different approach than Hyperbolic/Schodringer/Wave-like PDEs

and even in the latetr one a dichotomia remains

[verbal nebula]

Our definition of viscosity doesn't apply to relativistic fluids

So in a sense the whole equation goes kaput

[astral vine]

So does the transport part of the equations which does not fit the definition.

(i.e. your remark is even more on point than initially claimed)

[verbal nebula]

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

This is interesting, I have very little training in Physics beyond the standard first-second year physics

Apparently people have tried to come up with a good way to get viscosity in relativistic cases

https://arxiv.org/pdf/2406.18434

[tall jolt]

are the Euler equations better behaved?

[buoyant pike]

Euler equations are worse behaved than navier-stokes

Viscosity acts like a heat equation regularization

[tall jolt]

how is this interpreted? Doesn't this suggest that there's something wrong with either Navier-Stokes or general relativity?

[buoyant pike]

No

It just means that in a relativistic setting, the assumptions made in deriving NS are not valid

[tall jolt]

by wrong I meant physically wrong btw

[buoyant pike]

?

[tall jolt]

just clarifying. But your two messages seem contradictory to me

[verbal nebula]

Yes, it suggests that Navier-Stokes equations are incompatible with 'physical' or 'real' fluids moving at relativistic speeds

Every mathematical model breaks down at some scale

It's just a question of at what scale, and why

I don't work in relativistic fluids at all. I barely understand Newtonian fluids as it is

That transport term in Navier-Stokes is messing me up

[astral vine]

A funny thing is that on some specific Pseudo-Riemannian manifold the Navier-Stokes equations are well-posed for free. The manifold is just a twisted version of Rn by some non-zero torsion (continuous) parameter. In which case it behaves somehow like the Heisenberg group (not really but close). When the parameter is 0 one obtains the standard euclidean structure of Rn.

According to some people this should suggest that probably Navier-Stokes equations is not a sufficiently good model by itself since it is not robust under "natural" change of geometry.

I don't know if I agree with the last take

[stray fox]

Do you know if these people have suggested any alternatives?

[astral vine]

Nonlinear partial differential equations on noncommutative Euclidean spaces by Edward MacDonald in Journal of Evolution Equations, 2024.

[buoyant pike]

crank research

[astral vine]

Dead ass or kidding ?

[buoyant pike]

jk

Maybe

[astral vine]

It'd better be.

Tbh, In terms of direct physical application studying arbitrary PDEs, like fluids, in the non-commutative setting has no meaning by itself.

However, it is, in my not so humble opinion, still quite interesting to see in which sense one can stress the behavior of the said PDE, and to derive other inner and subsequent behavior.

Etc.

[neat spoke]

Is this a floor or a ceiling?

[tall jolt]

I think this is the same as Evans theorem 5.6.6? In that case it's floor (see the proof)

and I would hope that the general consensus is that [x] is always floor

[neat spoke]

Could someone explain this inequality to me?

This is theorem 6 on appendix C.4 of Evans' book

[verbal nebula]

Sure, $V \subset \subset U$. In particular $V \subset U$. Integration over a larger set is larger. Finally, x and y are independent from one another. So you can use Fubini's to swap the order of integration. Now the integral of a mollifier over it's ball will be 1

[untold delta]

MoonBears-C-

[verbal nebula]

(note fubini-tonelli might need to be checked precisely, but it'll all work out)

[neat spoke]

Why are we integrating over a potentially larger set tho?

What's the point?

[verbal nebula]

What are you trying to prove?

[neat spoke]

Wouldn't we get a similar estimate by integrating over V itself?

[verbal nebula]

Look at inequality (4)

[neat spoke]

Theorem 6 (iv)

[verbal nebula]

What is the statement of inequality (4)?

[verbal nebula]

The difference is that one is a limit, whereas what you showed is an inequality

[neat spoke]

Ye, but this inequality is part of the larger goal of showing this limit

My question is about why this inequality in particular is useful in proving this limit

More precisely

Why do we need to pick this larger W in the inequality

This is the whole proof

It's a technicality sure, but I am just curious about why is this technicality necessary

[verbal nebula]

Oh I see your question now. It's because as $\varepsilon \to 0$, it could be the case that it converges to some $f$ which blows up on the boundary. But since this is done locally, over compact sets you can always fit another compact set that stays away from the boundary of $U$.

[untold delta]

MoonBears-C-

[verbal nebula]

It's used obviously when passing through the limit from $V$ to $W$. It's just staying away from the boundary a little bit. You could probably replace $W$ with $\overline{V}$

[untold delta]

MoonBears-C-

[verbal nebula]

As long as it satisfies the same property that it's compactly contained in $U$. My intuition for this is coming from analytic functions on the disk that can't be extended to the boundary of the disk

[untold delta]

MoonBears-C-

[verbal nebula]

TL:DR it just makes sure that you're staying away from the boundary of $U$, where there can be some convergence issues

[untold delta]

MoonBears-C-

[broken hamlet]

Say $P$ is a quasi-linear elliptic operator, and we know that
$$\pdv{u}{t} = P(u)$$
has a solution for all time $t$. Now say $Q$ is a smooth function, can we deduce that
$$\pdv{u}{t} = P(u) + \lambda Q(u)$$
has a solution for all time (perhaps for sufficiently small $\lambda$)?

[untold delta]

shingtaklam1324

[quick pagoda]

I mean, the first such Q I might think of is Q(u)=u

Do we always have eigenvectors (not that we’re linear but close enough) of arbitrarily small eigenvalue? What about negative lambda

[median forum]

Need to go through the details but for Q(u) = u^2 there should always be problems with blowup for any positive lambda

[quick pagoda]

If not just u^2 some similar things should cause problems

[rotund jetty]

yeah you could just assume P(u) = -au for some a > 0, so this becomes an ODE. Or taking $P(u) = \Delta u$ gives the heat equation.

[untold delta]

L

[sweet osprey]

are Green's functions (for Laplace, if that helps) ever monotone in a way that depends on the boundary? e.g. if U subset V then G_U < G_V in U?

sorry if this is a silly question, I'm not at all familiar with PDE theory- was just trying to understand this estimate in Jerison & Kenig

[hidden coral]

assuming ur laplacian is negative this seems true, since you can consider G_U - G_V and apply maximum principle

[sweet osprey]

hm how does the maximum principle work here if the domains are different?

[quick pagoda]

maybe consider that difference on one of the domains? Idk if it works tho

[hidden coral]

If you have closure of $U$ is contained in $V$, then considering $F(x, y) = G_V(x, y) - G_U(x, y)$, by strong maximum principle we have $G_V(x, y) < 0$ for $y \in U$, and so thus $F(x, y)$ satisfies $\Delta_y F = 0$, and $F(x, y) < 0$ for $y \in \partial U$, so you can apply maximum principle to this again.

[untold delta]

Alphyte

[hidden coral]

maybe i made a sign error somewhere

[sweet osprey]

hmm
why can we use the maximum principle on just $G_V$? that's not harmonic by itself, is it? and why must $F<0$ in $\partial U$?

[untold delta]

bruhh

[hidden coral]

F must be strictly negative on the interior of V because it's a (strict) sub solution to -laplace = 0

[sweet osprey]

how does that work
F + const also has that property right?

[hidden coral]

the value on the boundary is 0

[sweet osprey]

on the boundary of U?

[hidden coral]

V

[sweet osprey]

F isn't even defined there though

because G_U isn't

[hidden coral]

I thought U \subseteq V

[sweet osprey]

yeah exactly

[hidden coral]

sorry I'm saying

G_V is strictly negative on boundary of U

so F is strictly negative on boundary of U and thus maximum principle dictates that F is strictly negative within U

[sweet osprey]

so youre saying uh
G_U is zero on dU?

[hidden coral]

that's the definition of greens function no?

[sweet osprey]

mb in my head i thought it was a contradiction bc the paper i read said wlog normalise it so |grad G| = something on the boundary
but grad G =/= G im stupid

[sweet osprey]

actually hgmm
this gives G_U leq G_V
but since they're both negative, this means |G_V|<|G_U|
and the paper i was looking at seems to claim the opposite result

nvm i can't add

[tall jolt]

what's a nice text treating in detail the short time theory of the Ricci flow (existence, uniqueness, regularity)?

[copper finch]

Since I stumbled upon a very good lecture (free on youtube) about elliptic differential operators, I thought I might share it here. It is not easily findable since it has so few clicks.

https://www.youtube.com/@ISem27

[quick pagoda]

There are also notes for ISem27 as well, should be available on one of the organizers’ pages iirc

[copper finch]

Oh you took this course?

oh actually it seems the course was held at my university

that is crazy

I found it by accident

[quick pagoda]

Not at your uni but

I believe they’re making a book of it? Someone else probably knows more on details

Isem28 is ergodic structure theory iirc, and 29 will be on eventual positivity

[copper finch]

Well if my exam goes well I could possibly apply for the next seminar

[quick pagoda]

Starts in about October

[astral vine]

This is very particular topic of elliptic differential operators theory : the Kato square Root Problem, which is not THAT popular amongs general PDE analysts, because too "abstract"

[astral vine]

I was for some reason invited to attend it by a kind of formal invitation I don't know why since I am a bit far from this approach of semigroup theory, while I am still aware about it.

[quick pagoda]

Iconic

Probably because connection with people involved

[astral vine]

Not that much actually

[quick pagoda]

Then I dunno, curious to invite then

Or maybe they wanted to get you to run a project

[astral vine]

I don't have time for this right now.

The problem is that if I run a project it will be about Semigroup theory, but nothing related to eventual positivity, because, it's kind of a generalisation of the maximum principle for the heat equation, at the abstract operator theoretic level, and it is not very handy in my opinion for the relevant PDEs (fluids)

[quick pagoda]

Yeah makes sense

They’re up to something but idk what

Fluids are spooky though

[buoyant pike]

Fluids are friendly

[astral vine]

Not that much on domains

[buoyant pike]

When the domain is a sphere

[quick pagoda]

Nothing is friendly on domains except maybe the laplacian

And even then maybe tenuous,,,,

[astral vine]

But it has no boundary, that's cheating

[quick pagoda]

Codimension 0 gg

[astral vine]

most second order Elliptic Operators are just different instance of the Laplacian

[quick pagoda]

They’re the laplacian in a trench coat

[buoyant pike]

Anatole do you know of a closed form convolution for the neumann to dirichlet operator

[astral vine]

On which kind of domain ?

[buoyant pike]

Ball hopefully

Circle maybe?

Something round

[astral vine]

Hmmm not really you can try to use, the Heat kernel on the ball ?

It is explicit

[buoyant pike]

Woah really

There's no closed form for the heat kernel on the sphere

Funky

[astral vine]

Okay not as explicit as you would probably like it to be, however you have point-wise kernel estimates

[buoyant pike]

Lol

What about point wise kernel values

[astral vine]

K_t(x,y) <= G_t(x,y)

K_t the kernel on the Ball

G_t Kernel on the whole space

[astral vine]

Like the explicit form is given by series of Spherical-like harmonics

[buoyant pike]

What is the correct basis for functions on the ball

Spherical harmonics tensor product r^n?

[astral vine]

Bessel-type function of r.

[buoyant pike]

woe

[astral vine]

and a power of r I think, do not remember clearly

[buoyant pike]

Whatever

This isn't even immediately important for my research

Just a group mate's

[astral vine]

'k

If I have time to look more clearly, I will get back to you but I am about to relocate my self in a new city etc.

[buoyant pike]

Yeah no rush

[astral vine]

So a bit buzy

[white idol]

yo guys

anyone here really good at writing PDEs

I need one.

[buoyant pike]

?

[neat spoke]

First silly question of the day, why is it that Evans says this identity only makes sense for functions which have regularity at least H¹_0?

And btw, why is the bilinear form defined on the left symmetric?

I am only having trouble with the middle term

I think it's just an integration by parts arguments, but there's a minus sign appearing in my calculation that shouldn't be there

[buoyant pike]

Well you take a derivative of u

And then integrate

So both u and the derivatives of u should be L^2

[neat spoke]

Why doesn't the expression also make sense for u in W¹,¹ tho?

Maybe this comes down to a more general question

Let $g$ be a fixed function in $L^{p}(U)$, where $U$ is an open (bounded ?) subset of $\mathbb{R}^{n}$ then the pairing:
$$
\langle f, g\rangle := \int_{U} |f(x)g(x)|dx
$$
is only well-defined for $f \in L^{q}(U)$, where $q$ is the Hölder conjugate of $p$.

[untold delta]

MisterSystem

[neat spoke]

?

If f is in L^q, then Hölder's inequality tells us the pairing makes sense

But what about the converse?

[neat spoke]

Ye I was tripping, this is not a symmetric bilinear form in general.

[rotund jetty]

he did not say that. He said "if only" instead of "only if". The reason for focusing on $H$ spaces is that their mathematics are much simpler than the $W^{k, p}$ spaces for $k \neq 2$.

[untold delta]

L

[quick pagoda]

Yeah since bounded by the Hölder inequality things, then dividing by norm of g?

Maybe?

Idk

Definitely if it worked for all g anyway

[neat spoke]

I am ESL, what is the difference between "if only" and "only if"?

Is he just saying "if u is in H¹_0, then the expression above makes sense"?

Instead of my other interpretation which was "this expression only makes sense when u is in H¹_0"?

[rotund jetty]

Yes. He says if only to stress that u being in H^1_0 is a weaker assumption than the previous assumption for u.

[neat spoke]

Oooooh

Ok

Thx

[neat spoke]

Why is |x| in W¹,infinity(R)? Doesn't it in particular imply that it should be an essentially bounded function? Which I am pretty sure it's not.

[quick pagoda]

yeah but its W^1,\infty_loc which is exactly locally lipschitz I think

something is a bit cooked to say its L^\infty tho

[astral vine]

your pairing is not bilinear

[neat spoke]

Oh yea

So forget the absolute value then

[neat spoke]

Could you guys give me an example of a bad domain U of R^n with non-C¹ boundary for which no sobolev extension operator E : W¹,p(U) -> W¹,p(R^n) say for every 1<=p<infinity (possibly including p = infinity?)

[heady silo]

try something like R^2 with positive x-axis removed

[quick pagoda]

Disk with slit but big

[heady silo]

hmm yes I remember the disk with slit as a counterexample for something but I dont remember what for, maybe it was exactly this

[quick pagoda]

W^1,\infty things

[neat spoke]

Smh why is W^1,infinity so different from the other cases

[quick pagoda]

but how different is it truly

[neat spoke]

You can't approximate by smooth functions...sad!

[quick pagoda]

True that is horrid

[neat spoke]

Omgomg

Something just occured to me

Let me write it

[heady silo]

disk with slit removed works as a counterexample for all p even infinity

[neat spoke]

A function $u \in W^{1,p}(U)$ is really an equivalence class of functions, so saying that $u \in W^{1,p}(U) \cap C(\overline{U})$ really means that there's some class representative of $u$ that can be extended to a continuous function on $\overline{U}$. But then, how is it possible to make sense of $u|_{\partial U}$ for $u \in W^{1,p}(U) \cap C(\overline{U})$? Because in principle, there could be many class representatives of $u$ which lie in $C(\overline{U})$ with possibly different values on the boundary.

[untold delta]

MisterSystem

[neat spoke]

But Evans defines a trace operator without ever dealing with this issue.

[hidden coral]

well perhaps you are interpreting this wrong

[neat spoke]

Same goes for the notion of support, what does it mean to say that Eu has support in V? Because the actual support, as a set, may depend on the representative of Eu. So is he saying that the support of representative of Eu lies in V?

[hidden coral]

u \in W^1,p \cap C(\bar U) should be interpreted as the continuous functions that are in W^1,p

[neat spoke]

Why isn't my interpretation valid tho?

Like, both seem to be valid interpretation

But one just makes the question trivial and the other not.

[hidden coral]

which question trivial?

[neat spoke]

Of what u|_partialU means

[hidden coral]

the trace operator should agree with restriction to the boundary for actually continuous functions

[quick pagoda]

If it’s discontinuous, then it’s less trivial

[hidden coral]

also, there is only a unique continuous representative of any L^p function

[quick pagoda]

(If one exists)

[neat spoke]

But when you interpret a continuous function as an element of W¹,p(U), what you are actually doing is considering it's equivalence class in W1p(U), right?

[neat spoke]

Ok

This is what I actually needed.

[hidden coral]

perhaps support can be interpreted as essential support here

[neat spoke]

Even so

[median forum]

Do you know why this is true? @neat spoke

The L^p thing I mean

[neat spoke]

Nope

[neat spoke]

Essential support of every representative?

Like

[hidden coral]

I mean essential support is defined in a way where changing the function on a set of measure zero does not change its essential support

[neat spoke]

The essential support of every representative has to lie in U?

[neat spoke]

Oh

[quick pagoda]

Highkey I’d say it’s more that consideration as an element of the space gives the integral/norm bounds for whatever purpose rather than caring about the quotienting

[hidden coral]

so the essential support of every representative is the same

[median forum]

Think about it. Suppose two continuous functions have the same L^p representative and see what you can deduce

[neat spoke]

I always forget the definition of essential support oof

[hidden coral]

i mean it's essentially the intersection of the actual support of every representative

[neat spoke]

Can you actually define the essential support like that?

[quick pagoda]

Anyhow, support within V just means it’s 0 off V

Don’t overthink highkey

[neat spoke]

In general I don't do it

But I really need to grasp the details this time

[quick pagoda]

Also remember equivalence in W^1,p implies a.e. equality but not the other way around

[quick pagoda]

The support thing can be just that f|{R^n\V} = 0

[hidden coral]

if u want to be pedantic you can say 0 ae

[celest dust]

Hi! I'm currently studying abstract PDEs. I would love to find some reference on evolution operators (the analogue to semigroups in non-autonomous problems) and exponential decay. It'd be great if this reference also treats these topics in scales of Banach spaces. Thank you very much!

[rotund jetty]

It just means $Eu(x) = 0$ for a.e. $x \notin V$

[untold delta]

L

[tired hollow]

Try Herbert Amann's "Linear and Quasilinear Parabolic Problems:, Vol. I: Abstract Linear Theory (Springer, 1995). It covers all three, evol operators and decay and banach. I am pretty sure its the most PDE friendly thing

Also see Veraar (2008) stochastic non-autonomous equations on UMD Banach spaces, assuming AT conditions; shows how interpolation spaces enter the fixed-point/regularity scheme. Good template for working on scales beyond Hilbert. You may want to read it sometime. It might fall extra to you but I like it alot; I used to work on it before

[celest dust]

Wow! Thank you very much for your answer. It was super useful!!

[astral vine]

In practice, I personnaly find Amann's book barely usable even if the collection of results is amazing.

Amann's style is something on its own.

This is obviously not up to date but Pazy's book is a more convenient introduction for evolution operators, but it does not fit the maximal regularity framework.

[tired hollow]

yeah but i feel like amann has an essense

yk what i mean

[celest dust]

Thanks, guys! I'm quite into the topic of evolution operators (of course I'm not an expert) but I need to check some results on exponential decay. I'll check all the references you gave me. Thanks again!!

[honest marlin]

Hello. I’m studying well-posedness of nonlinear parabolic PDEs in Sobolev spaces. Could someone explain the key steps (assumptions + a priori estimates + compactness/monotonicity tools) to prove existence and uniqueness of weak solutions?

[untold delta]

Henry Whitmore

[honest marlin]

1. How do we set up the weak formulation and energy inequality?
2. Which compactness/monotonicity arguments guarantee existence?
3. How is uniqueness proved?

[quick pagoda]

Well, do you remember how you can pull out the energy thing for the more boring heat equation setting? What happens if you just try to pretend it works here

[rancid crow]

Anyone knows a well written book on stochastic pde ? Ideally quite short and application focused

[astral vine]

It is possible if you have already the appropriate knowledge on Stochastic Analysis.

Such book usually does not contain proper introduction to stochastic Analysis (not for this I know at least)

[rancid crow]

I think I am familiar enough with stochastic analysis, otherwise I will catch up on certain things if needed

[tired hollow]

"Pao-Liu Chow — Stochastic Partial Differential Equations" is a application based

"Prévôt & Röckner — A Concise Course on SPDE" is concise

i mean tbh there are alotta good ones

ive done these two so i can comment

[rancid crow]

thanks

[astral vine]

I forgot to answer by then Martin Hairer's notes online are very good, from my pov.

But most SPDE lectures are only L²- based, which is sad since one needs Lp theory if one aims for sharp results. But this requires way more than standard Stochastic Analysis.

[warped star]

Hello, everyone

[buoyant pike]

Hello Johnson_23

[tired hollow]

How are you doing feller Johnson_23

[surreal epoch]

Checking the work of Roland Schnaubelt might be a good idea, especially his two surveys.

[clear sandal]

Anyone knows any good lecture notes on ODE theory mostly dealing with uniqueness and existence proofs?

[astral vine]

Viorel Barbu's book

[surreal epoch]

Hi, anyone knows a good ''new'' survey about admissibility of unbounded control/observation operators please?

[frosty haven]

not sure how much ODE theory you need, but some basic material can be found in chapter 12 of davidson and donsig (either edition)

[tired hollow]

bruv tbh i don think theres a broad, up to date survey devoted purely to admissibility of unbounded control/observation operators on the level of ever since Jacob n Partington

i cant really put my finger on it but ig tucsnak has a good one but thats also like 2013 i think

eh i dont think you want this but try

https://hal.science/hal-01095677/document

[turbid scaffold]

I like Theory of differential equations by Kelley

[arctic whale]

Is there a difference quotients characterization of H1,2 functions on Riemannian manifolds like the one we have in Euclidean space?

[rotund jetty]

They apply locally

For compact M, a function is in H^1(M) iff it is in H^1(U) for every chart U

[surreal epoch]

Thank you very much! Actually, I am reading both and I was hopping for a new one that covers recent results.

[astral vine]

I think his question tied to a more single-handed formula involving the geodesic distance d(x,y) instead of the eucldiean norm of the difference |x-y|.

[thorn furnace]

hello , anyone here did their phd , dissertation in PDE????

[buoyant pike]

Many people yes

[verbal nebula]

I'm doing my dissertation in PDEs

[copper finch]

This is from evans and I have problems understanding it. I am not the best with inequalities for integral though.

So like first of all I think the first inequality is just zeta being 1 on that part and gamma just rewritten kind of. (This is page 264)

After that we just have FTC I think.
Then its just product rule.

But the last part is weird to me. Can someone explain it to me? I mean like explicitely writing out the constant here

[buoyant pike]

Are you familiar with the statement of young's inequality

[copper finch]

well kind of. It is stated in the appendix. I think we only use it for the second term (p |u|^p-1 etc.) and rewrite it a bit but not sure. It looks just like the product in the young inequality but with the p multiplied to the other side kind of

[astral vine]

You can put the p in the constant C if necessary, it does NOT* play an important role here.

[neat spoke]

Could you guys recommend me some good references on fractional sobolev spaces?

[rotund jetty]

M. Taylor's books cover them

[astral vine]

He is mostly restricted to L²-type Sobolev spaces for the study of the fractional case.

[copper finch]

but is it right that the inequality is used on the second term or are they doing something with the sum I don't see

[astral vine]

Yes it is on the second product

[copper finch]

Thanks! Now I am pretty sure I get it

[sand echo]

for the sobolev slobodeckij spaces I think Leoni's book is the most comprehensive

[rotund jetty]

I believe volume 3 covers fractional L^p Sobolev spaces

[astral vine]

The definition is given yes, but as far as I remember not that much properties are given beyond the really basic ones, and in actual PDE applications of the book those spaces are only mentioned say twice.

Otherwise everything is Hs =Hs,2

[quick pagoda]

Iirc Leoni has a book exactly on this, I also have a book by Runst on these things lying on my office desk but it’s a bit older and quite odd

I forget if Leoni does the W^s,p or H^s,p-ish versions, I could check boy I’m lazy vro

Runst & (coauthor I can’t remember, I’ll check later) covers mostly the Besov/Triebel-Lizorkin-pilled stuff, but there’s so and so equalities so

Except at the integer values iirc? But the goal was non-integer ig

Triebel has his function spaces and wavelets on domains book too iirc, but idr what’s in it since it’s just sitting on my to-read pile in the office

Grafakos iirc talks a bit about them a lil in the classical/modern fourier book pair, but idr if he does much with them besides just defining by interpolation for W and probably a bit of discussion of H^s,p for p=2, maybe 1\leq p \leq 2?

So idk about having much in it, but it’s likely got a little bit?

[astral vine]

Only Wsp

[quick pagoda]

Gotcha, I just throw them all in as “it’s Sobolev something” mentally in remembering what’s where

[astral vine]

He is the kind of person, as a lot of others, that try mostly to avoid "abstract norms" and get rid of harmonic/functional analysis to define function spaces. But then all those people are then restricted to Sobolev Soblodeckij, which are in fact not so natural

(actually both scales are natural but, in terms of PDEs things in the whole space Hsp are more natural to reproduce more techniques introduced in the case p=2)

[astral vine]

Theory of function spaces I is readable also is II. But I won't recommend it for beginners. neither do I for Runst and Sickel.

Big Stein on Harmonic Analysis contains everything but since this is not the central aspect of the book, the results are then sparsely written throughout the book.

[astral vine]

Grafakos does everything. But his construction is rather designed for pure Euclidean Harmonic analysis people rather than PDE analysts.

[quick pagoda]

I didn’t mean the ToFS volumes 1-4 or whatever but he has a book specifically for domains, but that’s the one I had literally on my desk

Didn’t know Stein had that though

I should maybe grab a file for it sometime

[astral vine]

As said it is hidden throughout the book.

[quick pagoda]

Right yeah, that is troublesome, but I suppose that’s just the nonsense I have on the mind oop

[astral vine]

Some definition (especially homogeneous function spaces) are going AGAINST the spirit of the study of PDEs could it be on domains or not.

[quick pagoda]

Yeah true, idk some great pde-pilled one then

[astral vine]

There are none in my opinion

[quick pagoda]

Typical,,,,

[astral vine]

Actually there is a research paper by Kalton Mayboroda and Mitrea (2007) but this is a high end research survey, not something to learn

[quick pagoda]

There’s probably stuff in the GRAND TOME that is the Mitrea geometric harmonic analysis brick

But also more harmonic analysis pilled than PDEs ofc

[astral vine]

No because those books are actually designed for experienced researchers in my opinion.

Like everything is mentioned quickly.

[quick pagoda]

No human is reading them in full regardless

[astral vine]

They make no effort for notation, which makes everything unreadable for people not familiar with their work. And look who said it.

[quick pagoda]

Man why can’t there just be nice expository works that are actually human readable about the fractional things and with an eye to PDEs

[astral vine]

This will be the point of the lecture I start in two weeks.

[quick pagoda]

Kick me the lecture notes frfr

[astral vine]

Centred on Besov spaces, but will contain the rest for comparisons

[quick pagoda]

Ooh fun

[astral vine]

Otherwise I have my PhD thesis but it mostly concerns homogeneous function spaces and it already assumes the knowledge of inhomogeneous ones.

[quick pagoda]

Still handy

[astral vine]

There are basic PDE applications related to geometric analysis

And incompressible Fluids

Not hard to find online.

[quick pagoda]

The magical google machine

[astral vine]

Don't be fooled it is fully written in English

[quick pagoda]

I will attempt to dig it up sometime

Is the giant fluid PDEs brick out yet

[astral vine]

Wdym ?

Ah soon

Finishing the intro, everything was paused due to personal duties etc.

Relocation, mostly

[astral vine]

This would be the pinnacle of my research up to now to be honest, but not so humble.

Could it also be my summum level shit as hell, this does not contradict my previous statement

[quick pagoda]

Eagerly awaiting

[quick pagoda]

Fair fair

[buoyant pike]

In a change of coordinates, the divergence is scaled by the jacobian determinant right

And the laplacian is scaled by the jacobian determinant squared?

[astral vine]

Not really depends where you stop the propagation of the Leibniz formula

[buoyant pike]

woe

[astral vine]

Because you might obtain lower order terms.

div ( JAtA–¹ D u)

J the Jacobian determinant A the Jacobian matrix

So it scales like A² if kept in divergence form

But if you expend you will obtain second order derivatives of the change of variables, and first order derivatives of the function you applied the Laplacian

[buoyant pike]

Oh wait laplacian

Ok

Ummmm

Is the divergence scaling correct

[astral vine]

Due to pullback formula you can absorb either A or J with Du

I actually didn't checked the computation but used the formulas for 1 year almost every day obtained by differential geometry methods

[buoyant pike]

Unfortunately

I don't know any differential geometry

[astral vine]

You wouldn't be the first. That's not a shame or anything.

But surprising that all those people dealing with free boundary-type problems are reproving by ands Piola-type identities, while it is a 2 lines consequence from differential geometry...

The loss of time is insane especially writing those extremely heavy appendices

[celest dust]

Hi, guys! I'm studying some PDEs topics and I came across an inequality of the following type: e^{Cmax_{1\leq i\leq n} a_i^{b_i}} sum_{i=1}^nC_ia_i<M. Do you know if there are techniques in order to obtain an inequality of the form max a_i< something. I thought of something related to Lambert W function, but I'm not quite sure this works well in this situation. Thank you very much!

Here, C, C_i (1\leq i\leq n) and M are constants

[astral vine]

Could you be more precise please like is M uniform with respect to a's, b's or C's ?

Which kind of (precise) context did you meet such an inequality, do a, b or C's have more precise forms ? Etc.

When it comes to PDEs what comes in mind for me is the Gronwall Lemma

[celest dust]

I mean, what I want to do right now has nothing to do with PDEs, I just want to know if there techniques to obtain Max a_i< something from that inequality, such as Lambert W function for xe^x=C type of equation, for instance

[celest dust]

Yes, M is uniform on those quantities

[opal nest]

Navier stokes is undecideabke according to a rather advanced framework i developed and pushed to absolute conceptual and mathematical exhaustion. I have a full proof here for any that would care to double check. If I can get some assistance publishing outside of arxiv I'll let you co author.

https://drive.google.com/file/d/1d9M5S0sg_RGl7cvO4FFemaT0PpGwsVR-/view?usp=drivesdk

Please don't dismiss it until you've actually opened it, a lot of work has gone into this.

[verbal nebula]

Do you have a proof of anything you claim?

[median forum]

I am slightly confused why the “result” is called an axiom

[deft loom]

Please stop with all these claims of solving high level / unsolved problems.

[rotund jetty]

Before you share too much here and someone steals your work, publish on arxiv. Then advertise your arxiv submission. This is the standard practice in the math/stats community.

[astral vine]

I read it, and like... this does not actually prove anything. There is not even an actual outlined strategy.

If it has one, then you should need a lot of extra work and details to make it appealing to other people from the Maths community. You need to be exceptionally more precise and detailed to convince people.

But as far as we know the only reasonable and promising piece up to now, has been achieved Hyong-Mwan Ri and still has to be checked : https://arxiv.org/abs/2508.19590

[tall pasture]

I am seeing a couple different definitions for semilinear pdes. either

1. all derivatives of u occur linearly with coefficients depending only on x.
   or
2. The highest order derivatives occur linearly and the coefficients of those terms depend only on x.

are these equivalent definitions? for context, I haven't taken differential equations since undergrad basic course and had to add advanced pdes at the last minute so I am relearning everything and I am very confused.

[buoyant pike]

(1) is linear

(2) is semilinear

[tall pasture]

1 is not linear unless u and all derivatives occur linearly

let's wait for someone who knows this

[median forum]

?

They are right. You can separate linear into linear with constant coefficients and linear with variable coefficients but they are both linear

[astral vine]

So Ange apparently you don't know PDEs, so sad.

[astral vine]

Joke aside, never be that much confident and disrespectful towards other people. I bet after such kind of remarks, you will have to wait for ages before getting an answer that actually fits your desires. That's good because apparently you do not deserve it.

[heady silo]

woah!!

[astral vine]

You didn't ask yourself first if a possible confusion could come from your description, so that this guy could be right or not.

[tall pasture]

uh,, ok I only meant it in an objective way. it kinda muddles the feed to respond with incorrect basic things in an advanced chat

[astral vine]

Be a bit more humble (and see who said it)

[tall pasture]

see what about them

first of all, if you mean look at their little discord profile, I have no provable info there to say anything about their skill or understanding of this topic. secondly, to respond that way in the first place suggests immediately that they aren't really trying to help me at all. so it gives troll to me. and then for you to say "I bet after such kind of remarks, you will have to wait for ages before getting an answer that actually fits your desires. " kind of implies to me that troll response like this are normal unless I do something to show that I "deserve" real help. smh

[astral vine]

Honorable roles (golden/yellow names) are for longstanding people with mostly huge and notable contributions for this community, especially in the channel they are active in.

And, in particular, among these people, and this you might not know, Ange, is probably one of the biggest if not the biggest, notable members.

People do not troll that much here.

This is not the spirit

Answer back "I think you might be wrong because ..."

[tall pasture]

but also, I really don't mean to offend.. I am just having trouble

[astral vine]

And elaborate

[astral vine]

Be careful with the choice of words then (not so easy)

[tall pasture]

ok thanks for taking the time to clarify all of that. I appreciate you.

[quick pagoda]

Me when u is the 0th derivative of u

[tall pasture]

with due respect, since I now see that you all have probably put in a lot of work to understand this topic.. I get caught on details of wording. these definitions are really messing with me

[astral vine]

Concerning an answer to your initial question and definition stated as there are I agree with Ange.

So either the author is missing something while writing the def. Or you did when reading it. Or you did get it right and wrote the def. wrong (with typos/missing stuff) in this place.

[tall pasture]

ok i’m reading McOwen PDE book and here is the definitions

is this the general way of looking at it or are there competing definitions I need to be aware of?

also, I have been putting the pde's in operator form to determine linearity because it makes more sense to me that way. it there anything wrong with that way of approaching it?

[median forum]

Sometimes you also call equations that are linear in the highest derivatives quasilinear without restriction on coefficients

[quaint herald]

Wdym by this? What else is there for the coefficients (in the linear combination of highest order derivatives) to depend on other than x and lower order derivatives?

[quaint herald]

This definition is indeed a pretty common one, but there are different conventions. See for example Evans, I prefer his choice fwiw.

[median forum]

Misread mb

[tall pasture]

ok thanks for your helps

[digital ibex]

Could someone explain how where the “-“ went between the two results?

[nova vault]

@digital ibex can you post the full image and then perhaps we can see which results you're talking about if you refer to them with equation numbers

because as of now i do not see an issue

[digital ibex]

These are the lecture notes im using

The screenshots are from pg. 12 & 52 in the pdf respectively

[nova vault]

what are the eqns we should look at

okay this is a lecture combined into several lectures and there are several unnumbered equations

[digital ibex]

Yeah…
They arent numbered so i kinda just have to go w/ the page numbers

I looked up derivations of N-S online to see how others explain it and wikipedia honestly made it very clear

https://en.wikipedia.org/wiki/Derivation_of_the_Navier–Stokes_equations#Cauchy_momentum_equation

[nova vault]

awesome @digital ibex

charlie also has a book i think on turbulwnt flows

[digital ibex]

Ill need to get then when i get a chance

[nova vault]

or sorry had

unfortunately he passed a few yrs ago

[digital ibex]

But yeah,
Maybe its just me being smooth-brained, but it feels like the lecture notes didn’t explain it clearly

[digital ibex]

Yeah he passed in ‘21 i believe
He was the advisor of my current fluids prof.

[nova vault]

or has, i don't the right thing to say

[digital ibex]

I get what you mean

[nova vault]

oh nice @digital ibex

[digital ibex]

Do the screenshots make sense to you at least?
Since im still kinda stumped on where the - disappeared to before looking at wikipedia

[nova vault]

yes i mean they're just the definition of the stress tensor

okay go by the images you sent, which result is the issue pg 11 screenshot

*11

[digital ibex]

Ok

Im completely fine w/ how we got the - grad p on pg. 11
Its just when i did the same expansion for the tensor I ended w/ - div sigma

[nova vault]

so there's issue then ?

sorry i mean so there's no issue then

[digital ibex]

The text has positive div sigma

Im trying to find why i & the text disagree

[nova vault]

nope

look at the term multiplying \sigmaa_11

if you taylor it

you get \sigma_11(x) (\delta x /2) - \sigma_11(x) ( - \delta x /2) so finally there is a ppositive sign

similarly others

[digital ibex]

Wait, sorry
I thought sigma(x + delta x/2) and sigma(x - delta x/2) are opposing each other

So i figured a - would be attach to one of them

[nova vault]

$ \sigma$ is a tensor

it has in general many components depending on the space you're in

in this case there are 3 independent components assuming rotational symmetry

which is an su(2) symmetry

[digital ibex]

Sooo directionality isn’t important when writing sigma in component form?

[nova vault]

it is important not in this case because this is isotropic fluid

but there are cases where it will be

for example something called nematic fluids

[digital ibex]

I see

So the mistake is on my end then

[nova vault]

or fluids in like say D = 7+1 (SO(8)) or 10+1 (Type IIB) or even general

[digital ibex]

The text used F_1 in its example but im using F_3 to stay consistent

This is mines

eq. (8) is just the z component if -grad p

[nova vault]

sorry this is a cubic symmetry my bad

i am sorry so this a dihedral symmetry its just that \sigma_{ij} is the same as \sigma_{ji}

[digital ibex]

(I just noticed they all that z in the denom, thats my mistake but that’s not the concern)

[nova vault]

but i dont think it changes my arguments

[digital ibex]

I literally said that in my write up but didn’t think about thatttttt

Wait sorry do you mind if i take a few mins to write it out, just to make sure I get a pos. result and not a -?

[nova vault]

sure

[digital ibex]

Textbook says\
\begin{align}
\delta F_1 &= \sigma_{11}\left(x + \frac{\delta x}{2}\right) - \sigma_{11}\left(x -\frac{\delta x}{2}\right)\
&\approx \delta x\partial_x \sigma_{11}
\end{align}

I say (because all forces should be directed towards center of the object)\
\begin{align}
\delta F_1 &= -\sigma_{11}\left(x + \frac{\delta x}{2}\right) + \sigma_{11}\left(x -\frac{\delta x}{2}\right)\
&\approx -\delta x\partial_x \sigma_{11}
\end{align}

[untold delta]

KySquared

[digital ibex]

@nova vault

I (currently) think this is correct as I am going off of the figure on pg. 11

The normals are pointing out so the forces are directed inwards

[nova vault]

okay let us do this simply

consider a 2d case

fluid in a 2d box

like a pool or something and we turn on the jet, i guess

[digital ibex]

Okay

[nova vault]

we onyl have two directions now x and y , so the stress tensor is a 2x2 antisymmetric matrix

now lets say we choose to calculate the force on a fluid element in the x direction

the flow is horizontal lets say from $-\hat{x} \to \hat{x}$

[untold delta]

ben

[nova vault]

not necessary, but for simplicity

[digital ibex]

Right

[nova vault]

now what is the force on some random fluid element at some point (x,y) with area say \delta A

inuitively we'd say okay, the stress should be positive on the negative side and negative on the possitive side if we assign the coordinate labels of the area element [-x/2,-y/2] x [x/2,y/2]

wait that is what you got

lemme pull up the book

[digital ibex]

😭
Lol don’t worry, take your time

[nova vault]

it's just a convention

basically cahrlie is saying that a fluid can deform anyhow

but now we consider the deformation as positive when they happen against the flow

so \sigma is force on the - side due to the flow in the + side

but that will also change the total force on the rhs so NS remains the same

[digital ibex]

So if flow is moving in + x hat, deformation is pos., and if its moving in - x hat deformation is neg.?

[nova vault]

yup

it's just a convention ultimately it doesn't matter, you're totally right in your calculations you attached

[digital ibex]

So i just have to use that convention & I’m good to go
Thank you so much!

[nova vault]

you'll learn later that the stress tensor in general d dim is O(d) symmetric

[digital ibex]

I saw a proof in this text showing it is symm

[nova vault]

and perhaps get used to just writing the expansions fr the eqn, because one of my papers was working on fluid flows in graphene bernal bilayer in 2+1d and i had to solve a biharmonic, which is literally impossible analytically, and i decided to perturbatively solve it hehe

[digital ibex]

I didnt go through it fully but from what I glanced, it seems if the stress tensor isn’t symm, then the velocity would -> inf.

[digital ibex]

Ooooo
Is the paper public? I’d love to read it

[nova vault]

okay i should clarify this

fr isotropic it is o(d) if the fluid is chiral it is so(d), \sigma - - > ISO(\sigma)=O(d) in genrak where Iso(\sigma)=O(d) is maximal

[nova vault]

yes it is a nature paper

jesus science lol

i forgot my own paper

[untold delta]

Andrew

Andrew

Andrew

Andrew

Andrew

Andrew

[nova vault]

they are diff velocities @unkempt wharf

one is euler and other is the wave velocity

the euler is the velocity at which the flux moves

no the other one is euler

the wave speed is just the usual wave packet velocity

yes

yes

exactly

we can also prv this intuitively

like you said the flux is f = u v, and then wrote the quasi form

now on some wordline

the flux density is carried by -u \partial_x v(u) if the wordline is x(t)

sorry just \partial_x v(u)

so that is the euler speed

and you're right they'll match fr a linear flux flow

lemme tex somethin

can i assume you're okay with geometry

and analysis