advanced-pdes

[high rose]

So Maxwell's equations have an integral form and a differential form - could fractional calculus allow us to define intermediate forms that somehow interpolate between the two?

[pliant osprey]

I am looking for connections between (real) polynomial roots and PDEs. Could someone tell me a reference or some keywords (like Theorem names) to look into this connection? I think there should be something in form of stability theorems.

[tired hollow]

I don’t know wym by connections to polynomial roots. Differential Galois Theory (used in algebraic geometry under the name D modules) studies the Galois Group of differential equations and can be used to conclude when integrals have closed form solutions etc. Or maybe you mean something along the lines of phase plots? In that case something like bifurcation theory might be a useful one for PDEs

[slender fulcrum]

there is something called the characteristic variety of a pseudo-differential equation which is basically the set where the symbol of the pseudodiff op vanishes, it gives you some information about the behaviour of the pseudodiff op. For differential operators the symbol is a polynomial so the characteristic variety is the zero set of a polynomial.

[river path]

See the following, and related / referenced papers.
https://arxiv.org/abs/2104.06921

Tao has a blogpost on this too.

[twilit rover]

whats the best way to think of this

when M is the Torus or R^n it's basically just the L^2 inner product

maybe thats the way to think of it on general M. That the pairing is such that for smooth functions it is the inner product on L^2(M)

and then there is this one

I guess I can think of it in the same way as an extension of the inner product pairing

[astral vine]

wdym

[astral vine]

L²(M) is always the natural pivot (reference) space

just use localization and charts

But duality is not only about considering inner products

[pliant osprey]

Thanks, this looks promising.

[pliant osprey]

Thanks. This is also very interesting!

[river path]

[slender fulcrum]

[twilit rover]

So does this mean $C_{0}^{\infty}(\Omega)$ is dense in $H^{-1}(\Omega)$.

[untold delta]

IlIIllIIIlllIIIIllll

[lilac barn]

The answer should be yes because H-1 functions can be characterised in terms of L2 functions where compactly supported smooth functions are dense. (I am not sure if this answers the "does this mean" part)

[twilit rover]

We have $H_{0}^{1}(\Omega) \subset L^2(\Omega) \subset H^{-1}(\Omega)$.

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

The second inclusion is automatically dense because the first one is and $H_0^1$ is reflexive.

so I guess that finishes the proof, since $C_{0}^{\infty}$ is dense in $H_{0}^{1}$

[untold delta]

IlIIllIIIlllIIIIllll

IlIIllIIIlllIIIIllll

[junior bloom]

What does $\nabla\times u$ mean in distributions? Like obviously we associate $\nabla u\to\forall\varphi\in\mathcal{S}(\mathbb{R}):\langle -\nabla\varphi,u\rangle$ but what if $u(x,t)\in\mathbb{R}^3$ and we were looking at the curl instead?

[untold delta]

teafortwo

[exotic void]

If you allow the curl being understood as 2-form in $R^3$ then I think you can use the definition of $k-current$.

[untold delta]

shiburin

[exotic void]

https://en.m.wikipedia.org/wiki/Current_(mathematics)

[river path]

There's an integration by parts formula for curls.

In particular, <curl u, v> = <u, curl v> when u is compactly supported. Note that we dont pick up a minus sign.
There's usually a boundary term, it vanishes here.

I should really do this calculation. I havent checked that curl is self adjoint.

The point being: curl of a distribution is just given by acting the distribution on the curl of the input. Ofc for such things the distribution must take C_c^infty(omega, R^3) to R^3.

Alright I did the calculation and it works.

[junior bloom]

And that inner product is the Euclidean R^3 inner product?

I'll check it out, it's so weird how bad I am at calculus considering I do PDEs.

I was thinking about currents actually, independently of the suggestion here. However I really didn't want to work with GMT tools (which is what currents really are even though the Wiki lies and suggests it's differential topology).

[river path]

It's the L^2 vector fields inner product, where you do the usual inner product on the 3 components and then take the C^3 inner product for those results.

<u, v> = sum_j=1^3 integral u_j conj(v_j)

(u_j is a component, not a derivative)

[junior bloom]

Oh yeah, duh. Thanks.

[astral vine]

its kinda weird considering you had like 6 lecture where it was mentioned and used many times

😏

[river path]

Yes that is what made me realize I had to do the calculation

I just took it on faith when sylvie said it didnt add a minus sign

And i already forget the boundary term too

Actually I think I know it

It should be <u x normal, v>_del Omega

[astral vine]

This is straightforward from

just replace u by u_j and v by v_l

then sum the whole thing

[river path]

Bluh

[tired axle]

What is known about the conditions for which a hyperbolic partial differential equation is Hamiltonian?

i can't really find much about this

[hoary slate]

if M is a compact Riemannian manifold, why is it that the trace of the identity operator on this manifold is equal to the area?

[exotic void]

Does anyone know how to prove this Poincare inequality? $$\Vert u - \beta\Vert_{L^1} \leq C(\beta, U) \Vert Du\Vert_{L^1}$$ where $\beta$ is just a constant.

[untold delta]

shiburin

[exotic void]

for simplicity lets say u is smooth and the domain U is very regular.

[twilit rover]

@exotic void Doesn't it fail for $u = \beta + 1$

[untold delta]

IlIIllIIIlllIIIIllll

[exotic void]

Ahh yes lets say both $u>\beta$ and $u<\beta$ has measure larger than some nonzero multiple of measure of $U$ instead

[untold delta]

shiburin

[exotic void]

and the Poincare constant $C$ depends on that multiple and domain $U$ only

[untold delta]

shiburin

[exotic void]

How could I prove this version?

[twilit rover]

@exotic void There are well known such inequalities when $\beta$ is the average of $u$

[untold delta]

IlIIllIIIlllIIIIllll

[exotic void]

I am aware of this but I have no idea how to deduce from there

[scarlet fog]

Is it possible for a PDE to be quasilinear and semilinear at the same time?

I think the above classification does not "classify" PDEs well, i.e., a PDE might fall into more than one of these four classes

This seems appropriate to me

[river path]

Yes, that is the point

well

semilinear is the stuff described there setminus linear

and quasilinear is the stuff described there setminus semilinear

and fully nonlinear is the stuff described there setminus quasilinear

[tranquil steppe]

I'm not sure set-subtracting is a good idea. When people use these terms, it's usually because they prove e.g. "regularity for quasilinear elliptic PDEs", and then in particular their results cover the semilinear case.

(or in any case, nobody would exclude the special easier case when stating their result)

[river path]

Yeah that's valid

But if someone tells you "this PDE is semilinear" you would not expect them to write down a linear pde

This is not super consistent

[solid flint]

Hello! Can you explain to me please why they have used the projection in Lemma 5.1.2? Shouldn't it work without projection?

[bitter hollow]

I'm a bit rusty on the notation but it does seem like you don't need a projection since the tangential derivative already projects into the tangent space

Often such compact notation for vector calculus can be confusing (esp. due to derivatives being operators) so sometimes it's worth explicitly working out an example.

[slender fulcrum]

im looking for interesting topics for a bachelor thesis somewhere within the convex hull of PDEs, harmonic/microlocal analysis, spectral theory, scattering theory. does anyone know any cool results in these areas which may be worthy of a bachelor thesis?

[astral vine]

Bahcelor are the tree first years years right ?

[slender fulcrum]

yes

[astral vine]

If you are familiar enough with Fourier Analysis, Fourier Transform, and Fourier Series (the distinction here is important), you may want to check non-linear Schrodinger Equation with periodic or quasi-periodic (smooth enough) coefficients by the mean of the Bloch Transform

I'm quite sure one can have an elementary introduction, even if I am not able to find one

every reference I know are very heavy one

[slender fulcrum]

can you give a reference anyway, maybe i know enough to understand it at least a little

[astral vine]

I don't want to be pedantic

But I think not, but I'm still going to give it to you

P.Kuchment's book

[slender fulcrum]

Floquet Theory for Partial Differential Equations ?

[astral vine]

yes

[slender fulcrum]

seems readable, just some algebraic topology stuff i dont know

[astral vine]

You could check at the beginning of M.A.Johnson, P.Noble, L.M.Rodrigues, K.Zumbrun - Nonlocalized modulation of periodic reaction diffusion [...] after the introduction where a straight and very concise introduction to the bloch transform (not for Schrodinger Equations but 1-D reaction diffusion) is done but in a different scope, references therein by the same authors give complementary informations.

[astral vine]

Good for you if have enough knowledge at the end of your third year to know what is going to miss you to get the book

[slender fulcrum]

thanks ill have a look!

[slender fulcrum]

well who knows maybe im underestimating the book

[astral vine]

I think but don't hope so

Would be nice if you can grab the major part of it

[slender fulcrum]

but it seems like its mostly functional analysis for which i already had multiple lectures

[astral vine]

To be honest, this is not standard Functional Analysis, but if you had multiple lectures on standard F.A. as you said, you should able to handle it fast enough

[scarlet fog]

I understand everything in the pic above

but I don't see how the implicit function theorem is being used below, i.e., why can it be applied?

[scarlet fog]

Also, a different question: once I have the characteristic curves for a given Cauchy problem (first order quasilinear PDE), how do I get the solution to the Cauchy problem?

[lilac barn]

We have a function x(x0,t). How can you make x0 the subject to write x0(x,t) if not via implicit function theorem?

[lilac barn]

I would direct you here: https://web.stanford.edu/class/math220a/handouts/firstorder.pdf They have a full derivation on how to go about deriving the characteristics and solving the equation. To simply answer your question, it depends, sometimes it's just amount to solving the simpler characteristics and plugging them in, and sometimes it's get complicated.

[scarlet fog]

I agree with you, but how is the hypothesis being satisfied?

[scarlet fog]

Thanks, that helps

[lilac barn]

Unless I am completely misremembering, let F(x, x₀, t) = x − u₀(x₀)t − x₀. Then F₁(x₀,t) = 0 − u₀'(x₀)t − 1. Since we want to apply this on the origin, we have F₁(0,0,0) = −1 ≠ 0 and so IFT applies, where F₁ means the partial wrt x₀.

[scarlet fog]

Thanks a lot! I was making an inadvertent error

[civic turret]

how do you get the energy for the isentropic compressible Euler system with adiabatic exponent > 1? I tried multiplying with u as in the incompressible case but can't seem to make any progress

[civic turret]

got it

[hallow hearth]

bro is this one problem?

[civic turret]

Yeah the derivation of the expression

[hallow hearth]

😵‍💫 i should look into a psych0ology major

[untold delta]

Mikahopff

[solid flint]

Hello everybody. Could you please explain to me or recommend me some references about the classification of systems of PDEs of the form $u_t=Au$ ($u$ is a vector valued function, hence it is a system) When $A$ is a second order linear operator please?

[untold delta]

Mikahopff

[astral vine]

Ouhabaz's book

it mainly deals with scalar type Heat equations

but system-like heat equation can be treated similarly

even u with values in a Hilbert space not only a finite dimensional one

u_t = Au with A second order elliptic and non-degenerate is nothing with at least L infty coefficients is nothing but "a Laplacian" morally

[surreal heron]

Hey everyone, I am moving this problem I need help with here for the time being. I'm about to go on a jog, but should be back in about ~30 minutes or so.
(PDEs)
Let $\Omega \subset \mathbb{R}^n$ be an open bounded set, and let $b \in \mathbb{R}^n$, $c \in \mathbb{R}$ be constants.
Show that if the problem:
\begin{align}
\Delta u + b \cdot \nabla u - c u &= f \text{ in } \Omega
u &= g \text{ on } \partial \Omega
\end{align}
has solution $u \in C^2(\Omega)\cap C(\overline{\Omega})$ then that solution is unique provided $c \geq 0$.
Give an example to show that the problem may have more than one solution if $c < 0$.

I just want to make sure I'm not crazy and this actually is solvable.
This is what I've done so far:
Assume we have two solutions $u_1, u_2$ and set $w = u_1 - u2$.
This yields:

\begin{align}
\Delta w + b \cdot \nabla w - c w &= 0 \text{ in } \Omega
u &= w \text{ on } \partial \Omega
\end{align}

Multiplying the first equation by $w$ and integrating over $\Omega$ yields:

\begin{align*}
0 &= \int{\Omega} w \Delta w + wb \cdot \nabla w - c w^2
&= \int{\Omega} \text{div}(w \nabla w) - |\nabla w|^2 + w b \cdot \nabla w - c w^2 \text{ by Green's Identities/ Formulas }
&= \int{\Omega} - |\nabla w|^2 + w b \cdot \nabla w - c w^2 \text{ by divergence theorem and boundary condition}
\end{align*}

Thus we end up with:

$$
\int{\Omega} w b \cdot \nabla w = \int{\Omega} |\nabla w|^2 + c w^2
$$

I am supposed to derive that this implies $w = 0$, and thus the solution is unique, provided $c \geq 0$.
I am stuck here, however.
I've tried some manipulations with $w b \cdot \nabla w$, but cannot make this term 0.
Any suggestions on what to do next here?

[astral vine]

that's why they are called Heat equation

[untold delta]

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

[surreal heron]

Oops hmm

rendered earlier, going to debug my latex

[astral vine]

where is the BC in your weak formulation (the thing involving g) ?

[surreal heron]

The only thing we are told about (g) is that it is (C(\Omega)) iirc

[untold delta]

Draxton

[astral vine]

Yes and you don't need more

You also have to assume that Omega is bounded but also with smooth boundary

[untold delta]

Draxton

[surreal heron]

Hmm kk thank you.
This is what I was trying to do

I was trying to show that (w) has to be zero.
We know that (w \in C(\overline{\Omega}) so it attains max and min somewhere. If both max and min are on boundary, then they are zero, hence (w) is zero. But I wasn't quite able to show it's zero if the max is attained in the interior

[untold delta]

Draxton

[astral vine]

This is solvable fo f in L², and g in H^{s}(boundary)

use Lax-Milgram

if you have uniqueness in L²

you necessarily have uniqueness in the space of continuous fucntions

[surreal heron]

Oh snap

Kk thank you.

My professor didn't cover Lax-Milgram or Sobolev spaces, so he expected us to solve it using some method like what I was trying above lol. What you said sounds more approachable

This problem is from a class I took already, so it doesn't affect my grade or anything like that. I just am preparing for my quals and want to make sure I can solve everything

I'll try your approach and will report back here if I run into any issues. Going to go for a jog rn though, so I'll ttyl. Thank you for your help 🙏

[astral vine]

You have to be careful while existence (solvability) requires some regularity assumptions on involved objects

uniqueness is just about algebraic computations, with IBP (or Green/Stokes/Ostrogradski formulaes if you prefer), and basic L² energy estimates

that's why there is no regularity assumption made on f,g and the boundary of Omega (non other than IBP formla is valid( because there exists open sets for which the green identity may fail to occur))

[surreal heron]

Ohh I see, that makes sense, thank you.

[astral vine]

what happen to your last left hand side ?

should be a sufficiently big hint I guess

[twilit rover]

how can i find the eigenfunctions of the laplace operator on the right iscoceles triangle that is half the unit square.

[waxen bobcat]

Preliminary google search yields the following paper: https://evoq-eval.siam.org/Portals/0/Publications/SIURO/Vol3/Understanding_Eigenstructure_Various_Triangles.pdf

See Section 4

[twilit rover]

yeah this seems good

[untold delta]

Draxton

[astral vine]

if w is real valued the term vanish

equal to minus itself

[surreal heron]

Hmm kk, thank you. In this case it should be real valued. When you say it vanishes, is that specifically for the weak formulation?

[astral vine]

not really

just

[untold delta]

Functionanatolysis

[astral vine]

Therefore

[untold delta]

Functionanatolysis

[astral vine]

this is true for all sufficiently nice w, and divergence free b, such that you cna make sense of all above terms

[surreal heron]

Omggg

I actually derived that myself the other day but just assumed I had done something wrong XD

Hurray, thank you 🙏

(I derived that it was equal to it's negative self, but not that it was zero). That makes sense c:

Thank you for your help, it's greatly appreciated 🙏

[surreal heron]

Do you have any tips for how to come up with a counter-example for c < 0?
Like for example, would it make sense to drop to the 1D case and set b = 0, f = 0, and c = -1, and do something like the following

[untold delta]

Draxton

[surreal heron]

Then we would have an infinite number of solutions, the only requirement being

[untold delta]

Draxton

[scarlet fog]

https://math.stackexchange.com/questions/4521455/solve-yu-x2-u-y2-uu-y-0-for-ux-y-if-u2y-y-3y

I'm new to PDEs, happy to hear any thoughts on my working! Thanks

[twilit rover]

Let $f(x) = |x|^{2 - n}$, defined on $\mathbb{R}^n$, $n \geq 3$. Is is true that the only way $v = \sum_{|\alpha| \leq N}c_{\alpha}\partial^{\alpha}f$ can be bounded near $0$ is if $v = 0$?

[untold delta]

IlIIllIIIlllIIIIllll

[tranquil steppe]

Note that f is homogeneous of degree 2-n, i.e. f(cx)=c^(2-n)f(x). Similarly, if |a|=k, then d_a f is homogeneous of degree 2-n-k.
This implies that the sum can be decomposed into homogeneous parts (according to |a|) and the highest-order part dominates. If it is bounded, it has to be zero.

[twilit rover]

@tranquil steppe That's nice

[merry rune]

Would anyone happen to know any useful purposes for harmonic functions or maximum principle that is related to cybersecurity?

[odd crane]

Do you have any reason to believe there is?

This sort of question could do with more context

[river path]

Maybe something via markov chains? Idk

Elliptic curves = elliptic operators

[bronze gate]

If you regard the amount of cybersecurity (?) as a Markov process, then there are connections to certain elliptical operators

[river path]

Ha!

[merry rune]

I ask because my research is on harmonic functions, their properties (specifically maximum principle), and applications. When I finish my degree in December I plan to enter the cybersecurity career field. I have been looking for some way to connect my research to my career field though.

@odd crane @bronze gate

[river path]

Yeah i think your best path there is to move through connections with probability theory

If there are other connections, they'll likely be niche enough that they won't help you much

[slender fulcrum]

not a direct application of harmonic functions but maybe you could model certain cybersecurity questions with PDEs

[forest cradle]

maybe discrete harmonic analysis?

[bronze gate]

Hmm thinking about it more you could model cybersecurity via a differential game

[merry rune]

Could you elaborate?

[bronze gate]

Given the cybersecurity C of a firm satisfies a stochastic differential equation, say $dS=m(a,b,t,S)dt+c(a,b,t,S)dW$ where $a$ and $b$ are the controls chosen by the hacker and the firm. The hacker chooses a stopping time $\tau$ at which to strike, and gains a reward (from stealing, eg.) J(S,tau,some other variables)

[untold delta]

Karatzas and Shreve fan

[river path]

That's definitely all standard terminology

Or at least, it's fluent to me and we learned from completely different sources

[bronze gate]

...

This is pretty standard

A lot of mathematicians have worked on stochastic differential games

Elliot, Karatzas, etc

Controls=optimal control, hackers=stopper in a two player stoch diff game, firm=controller in a two player stoch diff game

I'm not using precise terms

I wouldn't use them if I was writing a paper

But this is a discord server, not arxiv

Oh okay 👍

[junior bloom]

If $u:\mathbb{R}^4\to\mathbb{R}^4$ what tools do I have to control the term $\lVert u\otimes u\rVert_{L^p(\mathbb{R}^4)}\lesssim \lVert u\rVert_{L^p(\mathbb{R}^4)$, where the 16- and 4-dimensional objects on the LHS and RHS respectively have their norm across dimensions taken in any standardly equivalent way for finite dimensional norms.

[untold delta]

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

[astral vine]

If it is the same p

there is no chance for it

unless u belongs to a ball in L infty

[astral vine]

Where does your question come from

what are the exact assumptions on u

It looks like

some sorts of non-linear estimates in Navier-Stokes

[junior bloom]

It's roughly the Navier-Stokes, I'm looking at dynamo theory stuff. What estimates did you have in mind?

I'm open to Besov or Triebel-Lizorkin spaces replacing L^p space.

[astral vine]

It depends what you want to prove exactly

I can do what ever kind of estimates you want

[junior bloom]

Nothing yet, I'm just looking for established theory

[astral vine]

Lp, and Besov are the "easiest" ones

Triebel-Lizorkin, only if standard Sobolev

[junior bloom]

Loosely speaking I'm looking for dominated controls on the terms in any nice space, then I'm going to review physical literature to see if the space is physically real.

[astral vine]

otherwise a pure Nightmare

Check out Bahouri - Chemin - Danchin

[junior bloom]

It's not actually Navier-Stokes but I want to adapt these controls because dynamo theory very similar.

[astral vine]

Mostly Lp; Sobolev;Besov control

the best space for navier stokes that "trivialise" the way estimates are done is

[untold delta]

Functionanatolysis

[astral vine]

Omega being a not toobad domain

and usually the whole or half space

[junior bloom]

We can take the trivial space where Omega is periodic.

Hm.

I'll check out BCD, neede a reminder that book existed.

Do you know if Triebel-Lizorkin continuously embeds into Besov spaces?

Ah I'm sure there are results about this in the standard harmonic texts.

Alright, thanks.

[astral vine]

Not all

Depending on integrabilities indexes

but cannot be used to prove nice enough results for Navier-Stokes or PDEs in general

[junior bloom]

Yes, it's quite dense. Check out Carmona and Delarue's 2-volume series on stochastic mean field games.

You would also be interested in Cucker and Smale's flocking/interaction kernel dynamics.

The technical machinery.

Heavy shades of ergodic theory and harmonic analysis if you're comfortable with these topics.

[bronze gate]

Probably very hard to do this

I don't do research in game theory, though I have seen it come up in specific contexts

So I can't give a definite answer

lol

one method used in game theory is what is called "backward SDEs"

[brittle belfry]

I'm a little bit over my head here, but I wanted to mention a book (looking at it right now) that seems to have content (and references) to pursuit/evasion, multiplayer games, etc. I'm seeing del operators, minimax examples...it might be too simple, but if it helps: Paul Nahin / Chases and Escapes : The Mathematics of Pursuit and Evasion

[junior bloom]

Backward SDEs are very general and in my experienced primarily used to resolve singular behavior, which I can imagine occurring in gamey contexts, but I doubly recommend what I previously did, on Cucker and Smale (Fields medalist) work in flocking dynamics.

Basically you have an interaction kernel describing how a bunch of agents evolve given information about each other, some dynamical environmental description, and environmental noise. What can we say about the system.

Like does it have solutions and how many, are there topological invariances, etc.

Are they stable or not, etc.

[bronze gate]

BSDEs originally found applications in stochastic control theory iirc (a lot of the problems become impossible to solve w/ hjb equations for eg, so you need bsdes)

[brittle belfry]

@tired hollow I...think I should've at'd you? I'm new on Discord. The book I just mentioned includes missile interception, guarding targets, cyclic pursuit, mobile/stationary evaders, hidden objects, etc. It also has copious references to source materials+

[twilit rover]

Is the optimal Poincare constant known for domains of the form $\Omega = {t_1v_1 + \dots + t_nv_n : t_1,\dots,t_n \geq 0, t_1 + \dots + t_n \leq 1}$.

[untold delta]

IlIIllIIIlllIIIIllll

[junior bloom]

@astral vine @quaint herald You two would be the best to ask about this. I'm trying to recall a tool from Fourier/harmonic analysis that lets me apply Calderon-Zygmund theory to attain boundedness for a singular integral operator based on its pseudodifferential operator symbol. My problem is when reviewing the relevant texts for some reason I'm only spotting results that require prior information about L^p boundedness or some other kind of finite difference regularity.

If you have a specific reference I would appreciate it.

Or maybe I'm recalling wrong.

I could swear there was technology like this but I didn't pay that much attention in harmonic analysis I confess

[junior bloom]

Yeah I think I remember the homogeneity of the pseudodifferential symbol being integral here.

[astral vine]

InHormander or Triebel books

You should have Exact statments

[solid flint]

Hello everybody, Do you know about any references about studying the local well-posedness of nonlinear PDEs/System of PDEs based on the well-posedness of the linearized one? ( mainly the continuity with respect to the initial date) Thank you in advance.

[astral vine]

this is too vague, what kind of system have you in mind

For shallow water waves like system like Saint-Venant and Green-Nagdi systems

https://perso.univ-rennes1.fr/vincent.duchene/files/CoursM2.pdf

(Based on linearization of some nonlinear Hyperbolic systems of first order)

[solid flint]

I don't have anything clear in my mind and I was wondering what kind of general/abstract results are there?

but I will appreciate it if you mention any other particular cases.

[astral vine]

I don't any general results

Or at least don't have in mind anything that "sounds" general to me

[solid flint]

any particular results?
Also, what were you exactly referring to in that documents please?

[shadow kelp]

What's the most readable introduction to microlocal/algebraic analysis that you know?

[astral vine]

AT the beginning using the non-linear Dirichlet to Neumann operator iirc

[merry rune]

As I can’t find a decent connective between harmonic functions and cybersecurity: what are some excellent applications (or extremely important problems) that are solved via harmonic functions and their properties? Any input is appreciated.

[merry rune]

Thank you.

[untold delta]

Hausdorff

Hausdorff

[scarlet fog]

Any thoughts, please?

[stray forum]

anyone help with these two problems plz?!

[astral vine]

is mainly about the use Fourier Plancherel for (i). (ii) is elementary, (iii) follows from (i) and a standard argument about densely defined bounded linear operators with value in Banach space.

[stray forum]

can you explain iii a bit

[astral vine]

I can't without giving the answer ...

[stray forum]

then give the answer please😩

[astral vine]

try before asking for the answer

[slender fulcrum]

if i understand correctly you have something of the form $$\lim_{\Delta t \to 0} \frac{1}{\Delta t}\int_{t}^{t + \Delta t} f(\sigma)\text{d}\sigma$$ where $f$ is at least continuous on $[0,\infty)$, so by Lebesgue's differentiation theorem you then get that limit exists almost everywhere and is equal to $f(t)$

[untold delta]

whzup

Hausdorff

Hausdorff

[scarlet fog]

plus we should use one of the other equations, I believe

[untold delta]

Hausdorff

[scarlet fog]

I think this works

[barren yarrow]

under which conditions does team B lose? so far I’ve two guesses:

• when all of team B is dead
• when A successfully made contact with B’s X objective

You haven’t mentioned if the individual team members have a certain line of sight (field of vision). Does an individual see another individual at a longer distance? if so how long of a distance

is the entire action taking place in 2d or 3d or does it matter?

also does each team have comms to allow them to talk to eachother? do they have spotters sitting at higher altitudes to see the entire theater or are they only relying on individual sight?

another question is if an individual dies, are his team-mates immediately aware of this fact?

i do believe it does affect the type of game. your initial description was very brief. i tried to help by asking questions so someone reading this can have a complete picture

it feels a bit like a modified soccer but with a maximum score of 1 because the game ends after the first “goal”.

but the both-die-on-contact condition i haven’t seen b4

@tired hollow have you considered writing some simulation.. maybe using a GANN and letting them duke it out and then replaying some longer matches to see what happened ?

are there no obstacles to the line of sight? if so, that removes the ability to hide which means everyone can see everyone else and it boils down to whether A has more members than B or not.

they may not have convergence guarantees but watching examples of games played by them can help, it can lead to an understanding of different strategies that emerge

i’m also wondering if deception as a larger theme is part of this game or not(this is related to my previous question about hiding)

[barren yarrow]

Most of the terminology in your message is unknown to me..

Then again I don’t know much game theory either

[merry rune]

I have found a relation for my research about harmonic functions that connect to cybersecurity (if anyone is interested). My advising professor has also asked me to delve into sub-harmonic functions and super-harmonic functions. Both of which I have not seen yet in any class and I’m kind of clueless about where to start on them other than a Google search. Any tips?

[white hazel]

I’d check out complex analysis books

[lilac barn]

They are both covered in Ahlfors complex analysis

[twilit rover]

Does anyone regret doing a PHD in pure maths, especially PDE

[lilac barn]

Not necessarily PDEs, but Qiaochu Yuan dropped out of his PhD in his fifth year, and he was doing in Pure Maths. The dude has 300k+ on MS and 100k+ on MO

https://qr.ae/pvUNzf

[astral vine]

About the PhD itself and math, not a single day in my life for 2 years now

But aside stuff could be very annoying

[white hazel]

I guess it could make you overqualified for some jobs but phds aren’t a hard commitment

Meaning you would probably drop out before regretting it

[twilit rover]

I am afraid of unemployment due to my bad performance

[twilit rover]

I'm considering applying to statistics only, but maybe I should apply to some math too

[white hazel]

Unemployment in academia is another story

Good luck cause it’s rough out there from what I heard 😮‍💨

[lilac barn]

Although it depends, not being able to get professor track doesn't usually mean you won't get employed in industry. One of my TA did his PhD in rational homotopy theory and published like a paper or two, pursued academia till postdoc, then quit, learned some finance and coding and now is employed at some hedge fund

[twilit rover]

I don't want to end up like the legendary Qiaochu Yuan

[white hazel]

Industry will always be way easier than academia

[lilac barn]

Tbh the case of Qiaochu Yuan is very unique because he got disillusioned with his own research like he said on his quora. Moreover, I think he is financially okay (rich parents according to his twitter) as well so employment wasn't really a concern for him.

[twilit rover]

even if my parents are rich, I don't want to be unemployed

[white hazel]

Just learn to code and you’ll be golden in the long run

[lilac barn]

I feel like these days getting into CS kind of jobs, especially math "heavy" fields like data science is not very hard. I know that companies like Google and Facebook do specifically hire math phds for their RnD departments

[twilit rover]

I studied CS, but CS internships are hard to get ... Mostly because I didn't get many interviews, and failed the one I got.

[white hazel]

Apply to a billion places

When I got my first internship I applied to like 200-300 places

[twilit rover]

yeah I appied a lot. I did get one though. Just not a "top" one salary wise.

[lilac barn]

Yeah, I think getting CS internships as a math undergrad or masters student wouldn't be easy because they would prefer having a CS masters or undergrad.

[twilit rover]

I am a CS and Math undergrad. I think my main weakness was behavioral interviews and talking about projects I did. I barely did any projects.

I was studying math instead of coding a webpage or mobile app

[lilac barn]

Yeah in undergrad, for CS internships they mostly rely on your portfolio and how many internships or projects you have already done. Especially when the applicant pool is filled up with students who have been coding for a long time, with internship experience with Google or whatnot

So I would say don't get discouraged by the result for now

[hallow pumice]

wait what happened to him?

[lilac barn]

He quit his Phd in his fifth year

[hallow pumice]

but what did you mean by "disillusioned with his research"

[lilac barn]

Like he thought his research was pointless and stuff

[hallow pumice]

oh he didnt quit because he was stuck of something?

[lilac barn]

No no, I doubt, the dude is insanely smart

[viscid sparrow]

This should be pretty obvious but my brain is dead.. Here is the question: Let $v \in W^{1, p}(\mathbb{R}^N \setminus \Omega)$ and $\overline{v}$ the trace of $v$ on $\partial \Omega$. Let
$$
w(x) := \begin{cases}
u(x) ,& \mbox{ if } x \in \Omega \
v(x) ,&\mbox{else.}
\end{cases}
$$
Show that if $\overline{u} = \overline{v}$, then $w \in W^{1, p}(\mathbb{R}^N).$

[untold delta]

Cookieman

[viscid sparrow]

I am able to show through divergence theorem: $\int_{\mathbb{R}^N} div(wf) = 0$ for all smooth function $f$

[untold delta]

Cookieman

[viscid sparrow]

Can we conclude anything at all about the weak differentiability of $w$?

[untold delta]

Cookieman

[viscid sparrow]

or am I just completely on the wrong track here (the reason I used divergence theorem of is because we showed the divergence theorem holds for traces in the same problem)

[lilac barn]

Is u in W^1,p(Omega)?

[viscid sparrow]

Yes

I am not sure where it went wrong... It's supposed to be obvious: We are gluing two sobolev functions together with boundary to be perfectly lined up.. The resultant function must be Sobolev..

[twilit rover]

@viscid sparrow Is it easy to show if $u, v \in C^1$?

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

the divergence of the product of two scalars fucntions ?*

[viscid sparrow]

w is scalar but $f$ is vector

[untold delta]

Cookieman

[astral vine]

I don't if your way could work

[viscid sparrow]

no it's not very clear that they are $C^1$. The only thing we have is $W^{1 ,p}$.

[untold delta]

Cookieman

[astral vine]

but should write instead

$$\int_{\mathbb{R}^N} w \partial_{x_k} \varphi = \int_{\Omega} w \partial_{x_k} \varphi + \int_{\Omega^c} w \partial_{x_k} \varphi$$

[untold delta]

Functionanatolysis

[astral vine]

then perform integration by parts on each integral

and see what happens

[viscid sparrow]

wait but I got $\int_{\mathbb{R}^N} w div(f) = 0$ from doing integration by parts on the right hand side

[untold delta]

Cookieman

[astral vine]

cannot be true

unless w is identically (a.e.) 0

ha and important thing to notice

If $\nu$ is the ouward unit vector of $\Omega$, then $-\nu$ is the outward unit vector of $\Omega^c$.

[untold delta]

Functionanatolysis

[astral vine]

and the boundary of Omega and Omega complementary are the same

[viscid sparrow]

Let me give it another try

[astral vine]

varphi is some Schwartz or smooth compactly supported scalar fucntion

[astral vine]

are ya winning @viscid sparrow ?

[viscid sparrow]

Yessir I don't know why I was struggling so much.. It is literally a definition proof with divergence theorem

[astral vine]

That's the whole thing of math

Seems difficult, but once you solved it, it's trivial

[scarlet fog]

Could someone help me with this non-linear beast? https://math.stackexchange.com/questions/4534503/solve-xu-x2-u-y2-uu-x-with-given-initial-conditions

[river path]

Is there a nice way to understand what exactly Besov spaces measure? I can see how they're capturing some kind of L^p mean Holder regularity, and I have seen a representation in terms of Littlewood Paley (which I should probably go look at again) but I'm not really sure how to see when they'd be more or less useful to work in.

[astral vine]

A nice one is teh heat demigroup formulation

It captures smoothness and decay at infty

Decay for the heat semigroup applied to your function -> regularity

regularity on your function implies decay/integrability on the heat solution associated with your fucntion as the initial data at time 0

For B^{s}_{p,q}

s is a pure regularity exponent

p is an itnegrability exponent

q is an integrability-regularity exponent

Check out Lemma 2.34 in the book of Bahouri, Chemin, Danchin

For the heat flow formulation of Besov norms

[river path]

That sounds very nice

Thanks

that probably gives a much better idea of when these are useful

i keep seeing them come up as natural spaces for estimates in turbulence theory

[astral vine]

That's somewhat how Chemin presented those spaces in few of its presentations during conferences, asking for the natural solution of the heat equation to belong to some Lq in time Lp in space

[astral vine]

Lq*(R+) is Lq(R+) with measure dt/t.

[sweet epoch]

I am reading an article that cites "Standard sobolev inequalities over the n-dimensional sphere", do you have any reference about it? I only saw sobolev embedding for open subsets of R^n

[river path]

i see i see

yes

[mild crown]

Any reccommendations for companions to Evans?

[river path]

Strichartz - distribution theory and fourier analysis is one of my favorite texts.

[mild crown]

is there a lot of good questions for the basic PDEs?

[astral vine]

I hope one day to write a Book about functions spaces and PDEs, that have Stein-Shakarchi and Rudin as requirements.

[obsidian wraith]

any suggestions on how to prove this affirmation?

[obsidian wraith]

should be any smooth function

well, at least C^2

[viscid sparrow]

I am a bit stuck on showing the following problem:

Let $\Omega$ be a bounded open set in $\mathbb{R}^N$ with smooth boundary. Let $A_n \in C^\infty(\bar{\Omega})$ be a $N$ by $N$ symmetric matrix with $v^TA_nv \geq \alpha|v|^2$ for all $v \in \mathbb{R}^N$. Define $T_n u = -\nabla u^T A_n \nabla u$ for $u \in H^1(\Omega)$ and let $\phi_n \in L^2(\Omega)$.

If $A_n \to A$ in $L^\infty$, $\phi_n \to \phi$ in $L^2$ and $u_n \in H^1_0$ are weak solutions to $T_n u_n = \phi_n$. Then show $u_n$ is Cauchy in $H^1$.

[untold delta]

Cookieman

[viscid sparrow]

I tried finding upper bounds for $|u_n - u_m|$ but nothing too useful came up..

[untold delta]

Cookieman

[astral vine]

It seems weird

doesnt T_n(u) =-div(A_n Nabla u) instead ?

[viscid sparrow]

Sorry.. it’s a typo. you are right.

[astral vine]

Use Lax Milgram and stuff to show that Tn and T are invertible. With appropriate bounds on the operator norms of the Inverse.

[astral vine]

For fixed phi

show that T_n^{-1}\phi converges to T^{-1}\phi

[obsidian wraith]

there is no entire problem, im trying to understand a line from a paper, thats all the info ive got

i can send you the paper but thats all thats used aparrently

there is a reference to some other paper that treats the latter as the second fundamental form of the gradient in the boundary

i dont know if im allowed to send files over the server but anyways

well from the rules i cant send paid articles apparently so ill delete it but i can send it privately

lemma 3.4 pg 332

the paper it refers to states the same argument

yes, the relevant one i forgot might be that c is positive

but the point is that it is referenced to a paper which does not posses the same hypothesis

which only asks for c to be in H^2 with normal derivative equal to zero on the boundary

[bitter yacht]

This is problem 6 in evans PDE on Harnack's inequality.
He says this is an explicit form of Harnack's inequality.
Can we from this inequality draw a conclusion on what the
constant in Harnack's inequality ought to be?
It should only depend on the geometry, and thus in the
case of the image, ought not to depend on x.
But I cannot seem to bound |x| appropriately so as to rid
myself of it in the expression

r^{n-2} ( \frac { r + |x| }{ (r - |x|)^{n-1} } )* u(0)

[white hazel]

I’m not sure if you can get a bound on the whole ball since e.g. you could have a singularity or something on the boundary

You can do it for any compact subset of the ball though

[bitter yacht]

@white hazel : yes I should have added that, my bad!
I did consider a compact subset B(0,R) of B(0,r), where R < r.
Yet I have failed in ridding myself of x and finding the constant
in terms of R. The fraction is a bit unwieldy and hard to control, I find

[white hazel]

Unless you want something like an optimal constant I think just doing the “obvious” stuff should work

Like replace |x| with r, R, or 0 to get bounds

[bitter yacht]

@white hazel : yes I should perhaps also add I am not looking for an optimal constant,
just an explicit constant depending on the geometry.
since x is in B(0,R) \subset B(0,r), I can bound |x| < R, and thus have R + |x| < 2R
But the denominator R - |x|, I do not have a lower bound for. All I know is that |x| < R
and thus 0 < R - |x|, and that zero is not helpful unfortunately. Am I overlooking something?

[white hazel]

For an upper bound you can bound the numerator with r + R

And the denominator with r - R I believe

[bitter yacht]

@white hazel: perfect! how did I not see that. thank you so much

Hm, this shows that sup u is bounded above by
r^{n-2} * (\frac { 2*r }{ (r - s)^{n-1} })*u(0).

Harnack's inequality reads: sup u <= C * inf u
inf u is bounded below by some other constant, so
I can't see how to read off the constant I am seeking.
Probably need to get some sleep, I think I am failing to
see a trivial step somewhere..

[white hazel]

Oh then that’s a little different

Basically at each point z in the compact ball, you can find another ball that sits inside B(0, r)

And you can use harnack’s inequality at each point but replace 0 with z

[bitter yacht]

Then we get, which Evans also states in his book, this

(1/C) u(z) <= u(x) <= C u(z)

something like that you had in mind? I tried with it, but
did not succeed with it either

I got another constant for the lower bound, which was not
the reciprocal of C that appears in the upper bound I got

[white hazel]

That’s okay

As long as you get constants on either side you can just find a large enough C so that it works

[bitter yacht]

Ah neat! In the first case I got
u(x) <= A1*u(z)
where A1 is an expression comprising r and R.
One condition I can impose is thus C >= A1.

In the other case I got A2*u(z) <= u(x),
where A2 also is an expression compsiring r and R.
The second imposable condition is 1/C <= A2, that is, C >= 1/A2

So I can simply take C >= max(A1, 1/A2)

I think that should do it. big thanks @white hazel

[grave oyster]

If $f(x,t_0)$ and I want to take the partial derivative $\frac{\partial}{\partial t} f(x,t_0)$ I would get zero yes? t_0 is some constant

[untold delta]

THAT'S MY QUANT, MY QUANTITATIVE
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tired ether]

Seems straightforward from the definition of a partial derivative

Also something better suited to #multivariable-calculus I think

[grave oyster]

hmmm, the reason I second guess because then the homework would be straight forward and near exactly what the lecture notes says

do you know much about interest rate theory?

[tired ether]

I don't, but maybe someone else does, so you can go ahead and post the problem in an appropriate channel

It could be a typo if it seems more trivial than you anticipated

[grave oyster]

what channel would be good for riccati equations

[mild crown]

unrelated but quant with big quads makes for a good funny, but yes just like the eigenvalue fella said

[round void]

Not sure if this might be more suited to #odes-and-pdes but I didnt manage to get an answer there... Does anyone know what do the "characteristic variables" mean in this context?
https://cdn.discordapp.com/attachments/908110794569449553/1022492552789753936/unknown.png

I know about the method of the characteristics but it usually applies to the transformaton of the independent variables right?

[tired ether]

I believe #odes-and-pdes ?

[solid flint]

Hello! I wonder how they get this energy formula? anyone has an idea please?

[frigid pilot]

i’d look into the physics of the problem and in particular look for a form which has physical constants restored in the equation, so you can do dimensional analysis. usually you can identify some kind of potential/kinetic energy terms

[ionic lodge]

Can someone explain how the big O is simplified here?

[unborn quiver]

This is better for #numerical-analysis, so I'm going to move over there, but what scheme are you using

[solid flint]

Thank you!

That's what I have tried at first, but it seems to be completely different here.
In the context of the wave equation, the energy is gotten by multiplying the equation by the velocity then performing some integrations/ Green's formule ..etc. ( a process that I never understood its soul) but here this doesn't seem to give the same results.

[viscid sparrow]

Are there any good complementary lecture notes/books for Evans? Some of the proofs on Evans is a bit too terse

[astral vine]

Depends on the topic. But generaly, a lecture more related with Fourier Analysis and Tempered (or general) Distributions is some what a great plus, F Golses' notes for instance. Another good complement is Brezis.

Otherwise any Book more specialised on several topics, like Fourier Analysis, Functional Analysis, Harmonic Analysis or Interpolation Theory, could be a huge plus. See C. Hao's lecture notes on Harmonic Analysis for instance. Ouhabaz's Book on sesquilinear forms and Semigroup (Analysis of the heat equation on domains). A guide to Spectral Theory for an introduction to linear PDEs via the formalism of unbounded operator on Banach spaces (more especially Hilbert). Alessandra Lunardi's Interpolation Theory for what it is named for and sectoriall operators and introductory Semigroup theory.

Many other références could be relevant depending on the kind of PDEs you want to investigate.

[viscid sparrow]

Thank you for the suggestions! Let me have look into them

[astral vine]

Maybe it is unclear but Brezis do not check distributions.

A badly formulated sentence.

Just another good complement for Evans. ( I would even say that Evans is a complement for Brezis...)

[verbal nebula]

I don't find chapter 4 of Evans to be very good, I preferred something out of Stein and Shakarchi's Fourier Analysis; also knowing your basic ODEs very well out of something like Boyce & DiPrima cannot be overstated

But as Anatole has said, it all depends on the "flavor" of PDEs/Analysis you're doing. For me I've found Han & Lin a great supplement, along with Gregory Eskin's book on Sobolev Spaces

Also at the end of each chapter, Evans provides a list of references for each chapter for each specific topic

If you're wandering through Chapter 2, it doesn't hurt to go back to Complex Analysis and look at the maximum principle there

If you're interested in the applied side of PDEs, a mathematical methods for physics/engineer book can be kind of interesting. Especially regarding Green's Functions

There are also interesting computational books where you talk about numerically approximating solutions to PDEs, etc.

PDEs is so vast, so find out what your interests are in Evans (because you can't cover it all) and dig in

[bitter hollow]

IMO the "interesting" part of Evans' book is Part III of the book (Nonlinear Theory); there are very few books that give a decent survey of the nonlinear theory the way that Evans attempts.

[astral vine]

A lot of tools and stuff exposed in Evans are too much hand-wavy/sloppy

Espescially, at the beginning where there is no clear functional analytic setting, except in few parts where it is explicitly stated to be L²/H1

Which is quite annoying with regard to modern PDE theory

(except few fields/subfields of PDEs where it could be relevant, but one would prefer other references imo)

[verbal nebula]

I can see this being an issue, but then again it's designed to be an intro and not bogged down in the details

I don't know what it's like in France, but here in California grad students usually aren't ready for that level of rigor

[astral vine]

But this does not adress the issue that changing the ambiant space could completely change the behavior of the PDE

like changing L² to Lp

p=/=2

some linear PDEs will have either more than one, or no solution in the Lp setting while being well posed in the L² one

[verbal nebula]

In the first few chapters Evans is very explicit about where things are, but doesn't mention if the space changes all of this stuff goes wrong

[astral vine]

That's my point

[verbal nebula]

I think? Profs are good at pointing that out when they teach PDEs

At least the ones I had were

🤷‍♂️

[astral vine]

Yeah, I personnally didn't have Evans as reference during my Master Degree so I can't tell

those things were also taught to me

[verbal nebula]

I jumped into second semester PDEs (Sobolev Spaces) w/out First and he didn't use Evans

I got Evans on my own and began working through it because it helped me understand my research better

[astral vine]

I bought because everyone were like " that's the standard intro book, a must have"

[verbal nebula]

But I do agree that Evans isn't exactly precise and this can give the wrong impression

[astral vine]

I already read whole Brezis before getting it

[verbal nebula]

My lower division prof took PDEs from Evans at Cal back in the hey-day, and I think that it's popularity is due to how it bridges the gap from the standard undergrad US curriculum

To the grad math experience

Rather than being a rigorous full steam ahead grad math book

[astral vine]

Yeah, I see

[verbal nebula]

So in that sense, it fully succeeds at conveying the "spirit" of grad PDEs

Without worrying so much about the technical details

But if you come at things from a functional analytic perspective, you will find Evans fully unsatisfying

Yet applied computational people don't focus too heavily on functional analysis

Which is a big audience for the Evans book as well

And you can see this in our taste for PDEs anatole, I am almost completely ignorant of the functional analytic approach

[astral vine]

Some heavy but standard Hilbert functional analysis

?

[verbal nebula]

I don't know any functional analysis

Beyond the baby basics

[astral vine]

To make converging schemes in L²/H1 , no ?

[verbal nebula]

I know L^2 convergence (roughly(

roughly*

[astral vine]

Oh okay, very interesting

[verbal nebula]

Fourier transform is some sort of isomorphism between l^2 and L^2

But my profs didn't focus on the functional analytic approach. I learned from an applied mathematician that did Machine Learning stuff, and I do research with someone in boolean fourier/harmonic analysis & probability

[astral vine]

I would like to see how you see like you solving Dirichlet Laplacian and stuff, with your knowledge for curiosity purpose

[verbal nebula]

I might have some time this winter break to delve into stuff ~ my main issue is I'm not up to speed on Probability w/ PDEs

I'll finish Han & Lin this december & January

Then Cafferelli's Fully Non-Linear Elliptic Equations

I'm hoping by spring/summer I can get into Figalli's Monge Ampere Eqns book

[astral vine]

Like Harmonic Analysis for Time Series ?

[astral vine]

Wow that's a huge program

[verbal nebula]

I can link you the work I've done Anatole w/ my prof

[quaint herald]

I think Evans is fine as a first PDE book. You get a taste of many different things and some intuition behind basic things like the behaviour of like parabolic vs hyperbolic vs elliptic pde is drilled in in a very accessible way. Indeed it skimps on stuff like distribution theory and abstract functional analysis, but there are plenty of other books to read alongside it, or to move on to as your interests develop.

Teaching a first course in PDE at most universities I would almost certainly choose Evans or Taylor as the main text, and then give a taste of some microlocal stuff by the end of the course.

[verbal nebula]

I'm of the same opinion minus the being qualified to teach PDEs part

[bitter hollow]

I disagree that it is handwavy at all, if you're referring to Part III. None of the results are sharp, but I do not think this is necessary to understand the flavor of the nonlinear theory.

The book does emphasize the study of behavior of solutions as opposed to studying PDE from a more general or functional analytic approach. This is intentional and I do not think it detracts from the survey of the field, with the possible exception if you are trying to focus on the linear theory.

[frigid pilot]

it’s impossible to focus on everything in a PDE book but I do think the lack of the fourier transform is borderline criminal

[tranquil steppe]

Well, Evans's book does cover the Fourier transform and mentions some of its applications, but it's just 10 pages.

I think this kind of treatment falls into "emphasizing the study of behavior of solutions", as TheMipchunk said.

[verbal nebula]

Most of Anatole's gripes are in Part I & II

[verbal nebula]

The fourier transform is in chapter 4 of Evans; it's not given a lot of detail because it's not a harmonic analysis nor fourier analysis book, it's a PDE book

[frigid pilot]

Yes but my point it's given very little attention, considering how useful it is in PDE. There's plenty of other PDE books like Folland which use it a lot more heavily. The fourier transform shouldn't just be limited to a harmonic analysis book, you can do a good chunk of chapter 2 using Fourier instead for instance. same with sobolev spaces

[verbal nebula]

and I think this is precisely part of Anatole's gripe

I like Evans for what it is, the exercises are clean, the exposition isn't too detailed

[bitter hollow]

Evans does explicitly avoid using the Fourier transform, it's true. I think he told me one time what that reason was, but I've forgotten. I can speculate since in my own research I very rarely use the Fourier transform: it is much harder to use for problems that have geometry (as opposed to on free space). In Part I where some classical formulas are derived for the laplace equation, heat equation, etc, the methods use (e.g. maximal principles) characterize the solution behavior in a very local way and thus often continue to be very applicable with complicated geometry, whereas with Fourier transform it is less clean.

[mild crown]

apologies to revive this thread but what makes you reccommend Boyce and DiPrima for fundamentals of ODEs? I had a rough time with Strogatz' ODE book because it would abstract away too often and the exercises seemed divorced to give you some reference as to where i'm at

[buoyant pike]

#odes-and-pdes

[twilit rover]

Is the pairing of $S(\mathbb{R}^n)'$ with $S(\mathbb{R}^n)$ jointly continuous

[untold delta]

IlIIllIIIlllIIIIllll

[frigid pilot]

by pairing i’m assuming you mean functional evaluation? ie if phi is a tempered distribution and f is a schwarz function then the map (phi,f) —> phi(f) is jointly continuous?

[twilit rover]

yes

I found that this is false

though it is jointly sequentially continuous

[ebon mesa]

Y'all got any good book recs for studying finite element methods?

[buoyant pike]

#numerical-analysis

[twilit rover]

How do I show that if $(e_j){j \in \mathbb{N}}$ is are the orthonormal eigenfunctions of $-\Delta$ on a smooth Riemannian manfiold $\overline{M}$ with boundary, then there is $C > 0$ such that for all $j \in \mathbb{N}$, $|e_j|{L^{\infty}} \leq C \omega_j^{n/2}$, where $\omega_j^2$ is the $j$th largest eigenvalue?

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

It's quite close to Sobolev embedding, but from Sobolev embedding and some further argument, I can only conclude that $|e_j|_{L^\infty} \leq C_s \omega_j^{s}$ for all $s > n/2$.

[untold delta]

IlIIllIIIlllIIIIllll

[slender fulcrum]

might not be that trivial, maybe have a look at this paper and its references https://www.ucl.ac.uk/~ucahalk/CharacterizationOfDefectMeasures.pdf

[twilit rover]

I found a proof in Hormander vol 3

[fierce nexus]

don't know if this is suitable for the advanced channel, but I don't understand the "insignificance" of the coefficients in the korteweg-de vries equation. i've seen the equation be represented with all coefficients being 1 as well, and i'm not sure why the change is not particularly significant

oh

never mind

i understand why the coefficients are not super significant, but i don't immediately see what substitutions would let us go from one set of coefficients to another

[past chasm]

need to also substitute in something like psi = a * phi

[twilit rover]

@orchid reef $\Delta$ is the Laplace operator

[untold delta]

IlIIllIIIlllIIIIllll

[astral vine]

Not on compact manifold without boundary

And usually with boundary this is Dirichlet BC

(at least this is usually assumed)

[twilit rover]

It acts on $H_0^1(\overline{M})$, so it is Dirichlet BC

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

It could act on larger spaces, but the eigenfunctions are elements of $H_0^1(\overline{M})$.

[untold delta]

IlIIllIIIlllIIIIllll

[fierce nexus]

wow yeah i see it now. thanks

[twilit rover]

Is it true that the norms $|u|{L^2} + |D^k u |{L^2}$ and $|u|_{H_k}$ on $H^k(\Omega)$ are equivalent?

[untold delta]

IlIIllIIIlllIIIIllll

[slender fulcrum]

i think thats a consequence of the gagliardo nirenberg inequality

[buoyant pike]

Yeah I think I saw something along those lines in my pde class

[twilit rover]

The proof I've seen is for R^n only. But there is one for bounded domains with a "correction term" https://en.wikipedia.org/wiki/Gagliardo–Nirenberg_interpolation_inequality#cite_note-17 that is enough to conclude the equivalence

I was trying to deduce the one for bounded domains from to the one for R^n using partition of unity, but the terms I don't want end up making the reduction difficult

[slender fulcrum]

wouldnt the natural way be to prove the R^n one from the bounded domains one?

i imagine there would be a lot of nasty terms from the cutoffs anyway

[twilit rover]

I think the natural way is to try to flatten the boundary, and then extend to R^n. But yeah the cutoffs introduce intermediate derivatives

[slender fulcrum]

alright good luck with that 😛

[hollow crypt]

it is significant if you want a nice Lax pair

[astral vine]

Yes

This due essentially to interpolation theory

You cna prove some of those inequalities by hand

but this some what limited

To show it simply on L², use Fourier Transform

on Rn

then use a well chosen Holder/Cauchy-Schwarz inequality on the Fourier side and get back on real life side via Fourier Plancherel

For Omega a lipschitz or a smooth enough domain, use your favorite extension operator from Omega to Rn

[fierce nexus]

i'm trying to understand the motivation for the KP equation coming from the 2d KdV equation. we start with $$u_t + uu_x + u_{xxx}=0$$ in the 2d case for $u:\mathbb{R}^2\to\mathbb{R}$ and the KP equation reads $$(u_t+uu_x+u_{xxx})x+u{yy}=0$$ for the 2 spacial dimensional and 1 temporal dimensional analog. i was wondering if anyone has any reasoning behind this

[untold delta]

maximo

[buoyant pike]

The motivation is to model nonlinear waves

Or are you asking about the individual terms?

[fierce nexus]

i'm just wondering how the KP equation is a bidimensional generalization of the KdV equation. i'm guessing i need more context to really understand why the KP equation is significant and tied to the KdV at all

i do see the obvious link between them, just not sure why del_x of the KdV + u_yy is particularly significant or at all motivated for a 2 spatial dimensional system

again im guessing it makes sense in some sort of physical context which i dont know of just yet

even further you have the three dimensional case $$(u_t + f(u)u_x + u_{xxx})x + u{yy} + u_{zz} = 0$$ which does have some semblance to the 2d case

[untold delta]

maximo

[buoyant pike]

Have you worked with water waves before

[fierce nexus]

not at length

[buoyant pike]

I see

Well it might be helpful to see to understand the motivation behind these equations

[versed shoal]

Probably a stupid q but I want to show that $$\int_{\mathbb R^d} \frac {\sin^2 ((y \cdot \xi)/2)} {|y|^{d + 2 s}} dy$$ is proportional to $|\xi|^{2s}$ (ie. is equal to $C_{s, d} |\xi|^{2s}$). This is quite easy in 1D, just substituting to divide through the $\xi$ factor inside the sine. How do you do it above 1D, does a similar thing work?

[untold delta]

George!

[astral vine]

just prove it for

[untold delta]

Functionanatolysis

[astral vine]

and this can be done via major play with Tempered Distribution theory : use the Gaussian, Fubini and the Dominated Convergence Theorem

[versed shoal]

is there a more direct way? I was thinking you'd be able to do a multivariable substitution of some kind

I've just never really had to come up with a non-trivial one

[astral vine]

Not as far as I know

For each proof I know that does not look in that "elementary way" : the shorter it is, the harder it is.

[versed shoal]

at risk of me looking lazy, could you elaborate what to do here?

[astral vine]

For $\phi\in\mathcal{S}(\mathbb{R}^d)$, show that

$\langle |\xi|^{2s}, \mathcal{F}\phi\rangle = \langle \frac{c_{s,d}}{ | y |^{d+2s}}, \phi\rangle$.

[untold delta]

Functionanatolysis

[astral vine]

to make appear the stuff, make appear a well chosen Gaussian and use the properties of teh Euler gamma Function

[lilac barn]

If you're comfortable with fourier transform, this integral would then be just the imaginary part of the Fourier transform of 1/|y|^{d+2s}

[astral vine]

Isnt what I said ?

[viscid sparrow]

Is there a nonhomogenous Dirichlet Boundary Data version of this?

I can't find any good reference to something like that..

[forest cradle]

tried Gilbarg Trudinger?

disclaimer: I don't do PDE and have forgotten most of what I once learned

[viscid sparrow]

Thanks for the suggestion. I found the result on there.

[quaint herald]

What range are you allowing for your $s$? Assuming you are taking $0 < s<1$ to guarantee integrability, then indeed it is as simple as in $1$-d. The integral is obviously rotationally invariant and then you have
$$g(t\xi)=\int_{\mathbb{R}^d} \frac{\sin^2(y\cdot t\xi)}{|y|^{d+2s}}, dy=\int_{\mathbb{R}^d}\frac{\sin(y\cdot \xi)}{|(y/t)|^{d+2s}}, \frac{dy}{t^d}\sim t^{2s}.$$

[untold delta]

gomez

[quaint herald]

And if you are considering some bigger range of s, entering distributional territory, then similar calculations will still work modulo whatever duality/regularisation you like to use to make sense of the integral.

[viscid sparrow]

In the proof, evans claimed the difference quotient $v$ is in $H^1_0$. I am not seeing why this is true.

[untold delta]

Cookieman

[viscid sparrow]

Especially why is the first term here equal to zero

on the trace?

[forest cradle]

the one thing I do remember from Gilbarg Trudinger was that it had many different versions of the same theorem for all sorts of similar but slightly different boundary conditions

[astral vine]

Using surjectivity of trace, the result trivially holds for any boundary condition g in H^{3/2} of the boundary

i.e. : for any g in H^{3/2} you can find (uniformly boundedly) a function v in H² such that its trace is g

so changing the unknown into w=u-v

F = f+Lv

you have to solve the Dirichlet problem

Lw = F with Dirichlet BC

[versed shoal]

0 < s < 1 yes

So you pick t = |xi|?

[versed shoal]

Wait is it true that Fourier transform preserves radial symmetry? If so then it fits together quite nicely

[quaint herald]

I mean there's different ways to write it, but yes one way is that once you have shown that $g(t\xi)=t^{2s}g(\xi)$ as I expained, you then have $g(\xi)=|\xi|^{2s}g(\xi/|\xi|)$ for any $\xi$, and the latter factor is a constant by radial symmetry.

[untold delta]

gomez

[quaint herald]

Yes, although you should be a little careful here as for your range of s, your integrand (excluding the sinusoid factor) is not locally integrable about 0, so not a distribution as written, and things like the Fourier transform are not defined by default.

But the argument for the radial symmetry of the Fourier transform of a radially symmetric function/tempered distribution is the same as the argument I would suggest for the radial symmetry of your integral.

Namely: $$g(A\xi)=\int_{\mathbb{R}^d} \frac{\sin^2(y\cdot A\xi)}{|y|^{d+2s}}, dy=\int_{\mathbb{R}^d}\frac{\sin^2(A^Ty\cdot \xi)}{|y|^{d+2s}}, dy=\int_{\mathbb{R}^d}\frac{\sin^2(y\cdot \xi)}{|Ay|^{d+2s}}, dy=g(\xi)$$ for orthogonal $A$.

[untold delta]

gomez

[versed shoal]

Got it cheers

[quaint herald]

np, happy to help

[viscid sparrow]

Which trace theorem are you using here?

I see now.. I've only learned $W^{1, p}$ trace theorem

[untold delta]

Cookieman

[viscid sparrow]

If a function is $C^2$ on the $C^3$ boundary of a open bounded connected set, can I extend it to a $W^{2, 2}$ function to the entire domain? This is some kind of trace theorem I feel like

[untold delta]

Cookieman

[viscid sparrow]

*The function is only defined on the boundary

[astral vine]

You don't really need sharper boundary

You can define it in an other way for higher regularity

[astral vine]

But yeah that's the spirit

[twilit rover]

How do you derive (6.9) on page 164 of Taylor's PDE book volume 1

I got the first equality, but I wonder about the second

Maybe I'll try to brute force the second

the first one I got by using formulas for graph coordinates

[still oxide]

Hello, I am looking for solutions in the form $u(x,t) = exp{i(kx-wt)}$ for the PDE $i\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} = 0$, and am trying to find the relationship between $w$ and $k$. I am unsure where to start this sort of problem (not for a graded assignment or anything like this), can anyone be of assistance on how I could get started?

[untold delta]

HaleyVinton

[charred bloom]

Plug u in your equation

[tired hollow]

May I know how I can self study this? Is there any video available to understand this?

[buoyant pike]

Finding a video for this will be difficult

What sort of background do you have

Have you seen integration by parts before

[tired hollow]

Yes, I have studied in Calc - 2.

[tired hollow]

I know Calculus, LA but not Real Analysis.

[buoyant pike]

Ok this is a very large leap

Do you know multivariable calculus

[tired hollow]

Yes, I do.

[buoyant pike]

And did you learn the divergence theorem/stokes theorem

[tired hollow]

Yes 🙂

[buoyant pike]

Ok so first you should read through a real analysis book

You can ask for recommendations in #book-recommendations

I would recommend revisiting this after some experience with real analysis

[tired hollow]

I understand your suggestion.

But I need to understand this for my exam.

[buoyant pike]

Unfortunate, I'm sorry for your loss

[twilit rover]

@tired hollow The main thing is the "integration by parts" formula $$\int_{\Omega}u_{x_i}v,dx = -\int_{\Omega}uv_{x_i},dx + \int_{\partial \Omega}uv N_{i},dS$$

All the other stuff is easy to obtain from this

Here $u, v\in C^{\infty}(\overline{\Omega})$

[untold delta]

IlIIllIIIlllIIIIllll

IlIIllIIIlllIIIIllll

[twilit rover]

This integration by parts formula can be deduced from the following generalization of the fundamental theorem of calculus: $$\int_{\Omega}u_{x_i},dx = \int_{\partial \Omega}uN_{i},dS$$

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

Which is proved in many places, e.g. https://en.wikipedia.org/wiki/Divergence_theorem

[tired hollow]

Thank you, mate 😊

[narrow ledge]

Is Brezis FA&PDEs too advanced to read along Real Analysis books like Rudin and Folland? Is the assumption that one already knows measure and hilbert space theory?

[buoyant pike]

Funny that this you should ask this right now

Paging @astral vine

[astral vine]

Only measure theory is assumed

Otherwise everything is made from "scratch"

I made a review of Brezis iirc

[narrow ledge]

(Lol I was gonna DM him but didn't want to bother him, since he studies about the same things I'm interested to study in the future I've asked him a lot of noobie things already.)

[astral vine]

#874143579096883240 message

here is the review

[buoyant pike]

(books prototype is not public yet)

[astral vine]

Well 'im going to send it by the mean of DMs

[lilac barn]

You kind of don't even need measure theory for half of the book, which is like Chapter 1,2,3,5,6. You will miss out on some context when you deal with examples, but nothing that you can't just comfortably take his word for the meanwhile.

He introduces the relevant measure theory in chapter 4, and gives you the bare minimum that you should know to get by the book. If you just cover that, or take those bare minimums as granted, you won't be missing out much, except some exercises which would be requiring you to use some technique from measure theory.

[astral vine]

While the content reintroduce some measure theory, few exercices require to be very well comfortable with measure theory, in order to be achieved without major troubles. That's why measure theory is still a requirement to me.

[tired hollow]

https://math.stackexchange.com/questions/2078957/method-of-characteristics-with-3-partials

Please explain me this step:

[lilac barn]

Differentiate the first equation using the second equation. Then solve the resulting equation

[tired hollow]

Yes, I understood most of the things.

[tired hollow]

I am struck here 🙂

[lilac barn]

Have you differentiated the first equation once?

[tired hollow]

I am sorry, I didn't understnad this.

If possible then would you please elaborate?

[lilac barn]

Calculate d^2x/ds^2

[buoyant pike]

I don't think they are having trouble with the first two equations

That is just a screenshot of the mse post

[lilac barn]

Ohhh

My bad

[tired hollow]

Haha! Yes

[tired hollow]

Here

[lilac barn]

They have just written the boundary condition, first in Cartesian coordinates, then in cylindrical then in the form of F they obtained

[tired hollow]

Yes, but would you please teach me how they derived this?

[lilac barn]

Just plug in z =0 in the "or in cylindrical coordinates" part of "An equivalent manner to express this relationship is"

[tired hollow]

Okay. Let me try 🙂

I am sorry, I didn't understand how they derived (pi/2 - theta) ?

[lilac barn]

Let p = tan-1(cost/sint). Then tanp = cost/sint. Thus sinp/cosp= cost/sint. As such 0= costcosp-sintsinp = cos(t+p). Then p = pi/2-t, (atleast in a pi neighborhood of t)

[tired hollow]

I see, Thank you 🙂

Now will you please teach me what they have done in last two steps:

[lilac barn]

First is just cos(pi/2-x) = sinx

The second just seems to be substitution

Plug F back into the equation for u they found

[tired hollow]

I see. Thank you, mate, for your help today 🙂

[lilac barn]

Yw!

[stark thunder]

I'm going to be honest, I think this is probably the place to post but not even sure.

I have a problem, I am trying to find all such g(x, u) (or at least classify some properties g must satisfy) where:

$\frac{t-49.5}{20} = \text{arg min}_u [ g(\frac{t^2}{20} - \frac{49.5}{10}t + 5, u) + p(t) u ]$

$\dot{p}(t) = - \frac{\partial}{\partial x} g(\frac{t^2}{20} - \frac{49.5}{10}t + 5, \frac{t - 49.5}{20}) + p(t) (\frac{t-49.5}{20})$

You can assume g is nice and there is at least exists one such g.
Is this a well-defined question? Do I need more data?

[untold delta]

Cursor

[stark thunder]

This comes from a more general problem I'm trying to solve in optimal control theory

[stark thunder]

I've refined it a bit

So I have a system:

$ p'(t) = - \partial_x g(x^(t), u^(t)) + p(t) f(x^(t), u^(t)) $
$ 0 = \partial_u g(x^* (t), u^(t)) + p(t) \partial_u f(x^(t), u^*(t)) $

$x^(t)$ and $u^(t)$ and f(x, u) are known functions related by the expression $\dot(x)^(t) = f(x^(t), u^*(t))$.

Find g(x, u) and p(t). Finding g(x, u) or making observations about g is much more important than making observations about p(t).

[untold delta]

Cursor

[buoyant pike]

Is this an Euler Lagrange equation

[noble bloom]

Hello, sorry I posted this in the other ode pde channel and wasn’t getting much response so I figured I might post it here

[spare root]

this is confusing lol

[next granite]

@noble bloom hi

[long crescent]

I am trying to follow the proof (in Hörmander) that $E = -|x|^{2-n}/(n-2)c_n$ is a fundamental solution to the Laplace equation for $R^n$ with $n>2$. I don’t understand the following step:

[\lim_{\epsilon \rightarrow 0} \int_{|x| = \epsilon} \langle \psi \grad E - E \grad \psi, x/|x| \rangle = \psi(0).]

I have been able to simplify the integral to

[\int \psi/|x|^2 dS-E(\epsilon)\int \sum \frac{x_j}{|x|} \frac{\partial \psi}{\partial x_j} dS]

But I am not sure how to finish or if this is the right idea.

[untold delta]

vivasvat

[buoyant pike]

What is psi

What is phi

[long crescent]

Sorry typo. psi is some smooth function with compact support (test function)

[buoyant pike]

So that should be psi(0) then?

[long crescent]

Yea

[buoyant pike]

What happens in polar coordinates

[long crescent]

So the volume element looks something like $R^{n-1} \prod \sin^{n-i}(\phi_i) d \phi_1 d\phi_2,…$ which is kinda messy

[untold delta]

vivasvat

[long crescent]

Ok so the right integral ends up looking something like $\frac{1}{(n-2)c_n} \int \sum x_j \frac{\partial \psi}{\partial x_j} \sin^{n-1} \phi_1 \sin^{n-2} \phi_2… \sin{phi_{n-1}}d \phi_1 d \phi_2, …$

But this just looks like I have made things more complicated

[untold delta]

vivasvat

[long crescent]

My intuition is that the left integral looks like it’s bounded by $O(\epsilon^{n-2})$. So the left might go to 0

[untold delta]

vivasvat

[buoyant pike]

Also have you used whichever greens' identity

[long crescent]

Green’s is used in a previous step. The right integral does almost look like the divergence of something but I’m not sure

Oh wait I used divergence theorem is that different then greens identity

[buoyant pike]

Yes they are different

[long crescent]

Oh

I think I already used it but I didn’t know it has a name

Actually I think I can figure it out, I had made some calculation errors above. I think I can get it to showing that $\lim_{\epsilon \rightarrow 0}\int_{|x|< \epsilon} \psi dS = \psi(0)$ this feels like a real analysis fact that I should be able to prove, but I am not sure how to do it.

[untold delta]

vivasvat

[long crescent]

Nvm figured it out, thank you for your help!

[long crescent]

Another question, for $E=(4 \pi t)^{-n/2} e^{-|x|^2/4t}$ , $t>0$ and $0$ if $t \leq 0$ (fundamental solution for the heat equation) why can we write

[\lim_{\epsilon \rightarrow 0} \int_{t > \epsilon} -E(x,t)(\partial\phi/\partial t + \Delta_x \phi) dx dt = \lim_{\epsilon \rightarrow 0} \int E(x, \epsilon) \phi(x, \epsilon) ]

(Where $\phi\in C^\infty_0$)

[untold delta]

vivasvat

[long crescent]

I think I figured it out just integration by parts.

[frigid pilot]

this is the answer to many many PDE questions lol

[odd crane]

I want to prove this basic Poincare inequality for $\Omega$, a square with side length $1$. I have to show that $$\left(\int_{\Omega}v^2 dx\right)^{1/2}\leq \left(\int_{\Omega}|\nabla v|^2 dx\right)^{1/2}$$ for every $v\in H^{1}{0}(\Omega)$. My initial idea is to use the gradient theorem, i.e. $u(x) = \int{a}^{x}\nabla u$ for some $a$ on the boundary (since then $u(a)=0$), but then I'm not sure how to relate this to the area.

[untold delta]

hOREP

[noble bloom]

I have this practice problem

I understand how to determine for what values of (x,y) it is elliptic, hyperbolic, and parabolic

but how do I use that given information to determine the characteristic vectors

[stark thunder]

No

It's coming from me trying to do something like inverse optimisation on the HJB equation (from optimal control theory)

Solutions to the backward equation will satisfy this constraint

[long crescent]

I am trying to find solution for the biharmonic equation in 3 dimensions. I know the answer is going to be the fundamental solution of the Laplace equation convolved with itself. But I don’t know how to calculate this. Another method that was suggested was the note the inverse Fourier transform is going to be radial (so the fundamental solution can be expressed as c|x|. I don’t know how to find what the constant should be equal to. Any advice would be appreciated.

[astral vine]

To check what is the constant you need to do Distribution theory to make appear the Euler Gamma and Beta functions

[gleaming lily]

y r u here

[river path]

To be more specific about what anatole said, try testing it against something. You know that for any sufficiently nice $\phi$, we have
[\int c|x| \Delta^2 \phi(x) dx = \int \delta_0(x) \phi(x) dx = \phi(0)]

[untold delta]

ryc

[river path]

Is there a nice phi where you can compute the integral on the left? Then c will be phi(0) / that integral.

Phi doesn't have to be compactly supported, schwartz (or even worse) is fine. But think schwartz.

[twilit rover]

How is the second term $\langle T, \nabla Z\rangle$.

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

oh

hm

[river path]

$Z^\ell_{;k}$ are the coefficients of the covariant derivative $\nabla Z$, which is a $(1, 1)$-tensor. To pair it with a $(2, 0)$-tensor means to first lower an index, which is what contracting $\ell$ against the metric $h_{j\ell}$ is doing, and then to sum over the two lower indices $j$ and $k$ against $T$.

[untold delta]

ryc

[river path]

I'm just explaining notation, idk if that's what you want or if you want something conceptual @twilit rover

The conceptual reason is "this is the directional derivative of Z in the direction T, but the coefficients of the derivative of Z are twisted by the metric for geometry reasons, so we have to untwist them first"

[buoyant pike]

Oh this isn't just the product rule?

[river path]

Yes, it's a covariant product rule

[buoyant pike]

Ok yes it is the product rule

[twilit rover]

no it's just the definition of the inner product

of tensors

well, more like it's a coordinate identity that follows from the definition

[twilit rover]

Why do the $u|{S_0}, Yu|{S_0}$ exist, and why does the Cauchy data vanish

[untold delta]

IlIIllIIIlllIIIIllll

[twilit rover]

this is page 505 of taylor's pde book vol 1

[low turret]

I’m interested in nonlinear PDEs with a compact time dimension (or, I suppose, time-periodic solutions of nonlinear PDEs. So, the PDEs are defined on R^n \times T^1 instead of R^n \times R. I want to be able to determine whether there is turbulent behavior for common equations like the nonlinear Schrödinger equation and the nonlinear Klein-Gordon equation (my intuition is that the “time travel” aspect of time-periodic solutions will lead to turbulent behavior across the system from nearly any perturbation), but also determine whether there is finite-time blow-up for otherwise well-behaved equations in the time-periodic case. I haven’t been able to find many references about similar research (I’m talking specifically about how to treat time-periodic PDEs) or thought of an approach to a proof.

[buoyant pike]

I believe that John Gibson has done some work on finding time periodic solutions to fluid equations

Numerically, of course

Tarek Elgindi has some work on the Euler equations where time periodicity shows up

And I believe the goal is to use these to show blow up for Euler equations

[astral vine]

A related area is about looking modulation for travelling waves in some pdes

leading to quasi-periodic/stationary solutions

Well thinking about it, I am not sure about that one

[buoyant pike]

Hello

What are the units on the heat kernel

The exponential has no units

And 1/sqrt(4pi*k*t) has units of 1/length

How does this become temperature

Oh well I guess you pick up the length*temperature in the convolution

[astral vine]

du/dt -μ∆u =0

μ is the constant from renormalized equation in m²/s

From the Fourier Law

Rescalling your solution will show up a √μ so that what is in the gaussian is dimensionless

Notice that you have to perform the rescalling on the initial data and the forcing term to

[twilit rover]

Why did he use Poincare? It seems unecessary?

[twilit rover]

well it becomes necessary in the following discussion

[stray forum]

Any hint will be appreciated.

[twilit rover]

@stray forum You can use the eigenfunction expansion of the solution? Or integrate by parts.

[stray forum]

this is unbdd domain can't do anything or please be precise in your answer.

[twilit rover]

it says open bounded and smooth

[astral vine]

First as @twilit rover mentioned, the open set is assumed to be bounded, with smooth boundary.

Moreover the theory of Sobolev spaces, and all general integration by parts formulaes are still true for (smooth) unbounded domains (even rough in fact), for free (do it for the half space R^n_+, then use charts, localisation via smooth partition of unity, on say Schwartz functions to ensure that the integral still make sense).

Finally, they ask you for uniqueness, not for existence

and your problem is linear.

[buoyant pike]

Have you tried energy methods

[stray forum]

thanks i will try in that way.

[fickle goblet]

I am dealing with calculus of variations for the first time, i have the functional:
$$I_\varepsilon[u]=\int_{\mathbb{R}^d}W[u(x)]+\frac{\varepsilon}{2}| |\nabla u| |^2dx$$
where $W\in C^2( \mathbb{R}^d,\mathbb{R}) $ and $ u \in C^1(\mathbb{R}^d)$ such that this integral is finite.
I have to derive the euler lagrange equations but I dont really know how to go at it, my main problem is the $\frac{d}{dx}\frac{\partial L}{\partial u'}$ part. I simply dont know how to take this deriavtive.

[untold delta]

Enoo58

[fickle goblet]

Also how is it justified to use $\nabla$ here since $u:\mathbb{R}^d\rightarrow \mathbb{R}^d$

[buoyant pike]

Ok yeah that's very bad notation and makes things unnecessarily difficult

[untold delta]

Enoo58

[fickle goblet]

yeah I thought so

[waxen bobcat]

You wrote $u\in C^1(\mathbb{R}^d)$ though, which usually means $u\colon\mathbb{R}^d\to\mathbb{R}$.

[untold delta]

Zanarcane

[fickle goblet]

yeah but i also plug it into W in the integral that wouldn't make sense if u is scalar

[buoyant pike]

Have you seen calculus of variations for 1d functions before

[fickle goblet]

not at all, we went straight into this

[buoyant pike]

I see

Ok so this notation is informed by what happens in the 1-d case

Where you typically have [I[u]=\int L(t,u,u')dt] and then $\pdv{L}{u'}$ is taken to mean the derivative in the third variable

[untold delta]

無名之輩

[fickle goblet]

yeah I read that online, does that mean i should treat u' as a variable and just take the derivative?

[buoyant pike]

Well you correct way to do this is to find a function $L(x,y,z)$ and then take the z derivative of this

[untold delta]

無名之輩

[fickle goblet]

in our case would that be the term $\frac{\varepsilon}{2}| |\nabla u| |^2$?

[untold delta]

Enoo58

[buoyant pike]

Right so in our case we are taking z to be nabla u

So what is the z derivative of eps/2*norm(z)^2

Well really it should be the z gradient of this

But also yeah there is something going on with the dimensions of this problem

And I don't think they match up

[fickle goblet]

eps*z?

[buoyant pike]

Yes

[fickle goblet]

yeah it doesnt make sense to me either like the other guy said notations suggest its R^d to R but then the W(u) term doesnt make sense

[buoyant pike]

Oh wait actually

So x in R^d

u: R^d to R

So W is a function of x and u so (R^d,R) to R

Maybe?

Anyways the notation is borked

[fickle goblet]

yeah I should write a mail to my professor to clear that up lmao

[waxen bobcat]

My guess is that W is an operator mapping functions in $C^1(\mathbb{R}^d)$ to $\mathbb{R}$.

[untold delta]

Zanarcane

[waxen bobcat]

And it is differentiable in the functional sense.

So, maybe $$\pdv{L}{u'}v=\left.\frac{d}{ds}L(u,\nabla{u}+sv)\right\vert_{s=0}=\varepsilon\langle\nabla{u},v\rangle.$$ But this is just my guess.

[untold delta]

Zanarcane

[waxen bobcat]

Then $$\frac{d}{dx}\left(\pdv{L}{u'}v\right)w=\varepsilon\left\lbrack\langle(\nabla^2{u})w,v\rangle+\langle\nabla{u},(Dv)w\rangle\right\rbrack.$$

Am I spewing nonsense?

[untold delta]

Zanarcane

[buoyant pike]

Why are you taking weak derivatives for all of these

[waxen bobcat]

I'm taking the directional (Gateâux) derivative, not necessarily weak derivatives in space, since I can't write the second term $\langle\nabla{u},(Dv)w\rangle$ in the form $\langle Aw,v\rangle$ for a matrix $A$ like the first one.

[untold delta]

Zanarcane

[waxen bobcat]

Or maybe just $$\frac{d}{dx}\left(\pdv{L}{u'}v\right)=(\nabla^2{u})v+(Dv)^{\top}\nabla{u}.$$

[untold delta]

Zanarcane

[tired hollow]

I've not taken a course in calculus of variation but i stumbled across this when reading bishop:
Could somebody explain how they got from 1.87 to 1.88? The book only mentioned using calculus of variation, but idk how.

oh nvm i googled it

[waxen bobcat]

Why did the integral over x vanish though?

[tired hollow]

https://math.stackexchange.com/questions/2130282/bishop-ml-and-pattern-recognition-calculus-of-variations-linear-regression-loss

defined it as a function G

and then took euler-lagrange

or well not wrt x, the intergral wrt t

[viscid sparrow]

What is a harmonic function that has positive outward normal derivative on the boundary?

on a $B(0, 1)$ ball in $R^d$

[untold delta]

Cookieman

[buoyant pike]

What have you tried

[viscid sparrow]

pretty much just $|x|^2$, which doesn't work unfortunately

[untold delta]

Cookieman

[buoyant pike]

Why doesn't it work

[viscid sparrow]

outward normal derivative is sometimes negative

I relax the condition.. I just need a function which has positive Laplacian and positive outward normal derivative

either way, $|x|^2$ still doesn't work..

[untold delta]

Cookieman

[astral vine]

just consider the appropriate neumann problem

[untold delta]

Functionanatolysis

[astral vine]

f negative, smooth up to the boundary

g non negative smooth on the unit sphere

just use the Poisson Kernel

this will give an exact formula for u in terms of f and g

this is straightforward

notice that this involves a compatibility condition

do you see any troubles ?

[viscid sparrow]

This is good. I think I found an example. Thank you!

[astral vine]

weird

[viscid sparrow]

What is weird?

[astral vine]

the compatibility condition could be written as

<f,1> (Omega) = <g,1> (Boundary of Omega)

what happens if f is negative and g non negative ?

[brave bay]

Why doesn't $|x|^2$ work? Hit it with $\Delta$ and you get $2n$, and its outward normal derivative is $2$ on $\partial B$...

[untold delta]

4453

[ashen drift]

can someone please help me get started with this

[buoyant pike]

What have you tried

This is Duhamel's principle right?

[ashen drift]

idk i havent tried anything, even looking at my notes its still pretty unclear

[buoyant pike]

Ok well try Duhamel first

[fringe charm]

In my physics class we’re doing retarded potentials and our prof just gave us green’s function but I wanted to actually derive it so I’m trying to find green’s function for the 3D inhomogeneous wave eq and then solve for the potential so I’ve done this so far

Should I change to canonical form and let $\xi = r+ct $ and $\eta = r-ct$ ?

[untold delta]

valdimirputin420

[viscid sparrow]

yeah I realized now I made a mistake in computing... Thankfully the proof to my problem required a weaker function than the solution to laplacian.. So I don't need this example anymore

[bold violet]

I have a question on greens functions that I'm just having trouble connecting by following the derivation

I'm sorta missing why when we look for greens function we look for solutions of a homogeneous problem

even if the beginning problem is inhomogeneous

I realize its probably super straightforward, just having trouble extracting it and it feels important

[astral vine]

Solve the Dirichlet/Neumann Laplacian on a Half space.

You can turn the problem into a vector valued ODE.

This may give some insights

[bold violet]

ah i can try

[bold violet]

you do mean like a half plane right?

[astral vine]

The problem is left unchanged

The dimension is not very important

Half space is for Rn+

Half plane is R2+

[bold violet]

hrm I may have asked this question in the wrong channel

but i will give it a go

I should maybe note ive never completed a greens function problem

I'm still just trying to understand it in general or the motions

im not sure how straightforward solving a problem in arbitrary dimensions is

[astral vine]

The insight for Boundary Layer operators comes from what happens when flatten the boundary

[bold violet]

darn now im almost certain I asked in the wrong channel

my apologies

i appreciate your help though

[astral vine]

?

[bold violet]

I should have probably asked in the normal ODE channel

[astral vine]

Assume the prescribed boundary value is the restriction of some nice C² function

Turn the problem into an inhomogeneous problem with 0 boundary value, with non zero forcing term.

[bold violet]

I appreciate your suggestion, I'll give it a go once I've done a few basic examples so I know the motions.

[fringe charm]

So can anyone help with this?

[tired hollow]

From my understanding the method learned to solve these is very new. Tried bringing it up to a diff DE prof. Said they’ve never seen Green’s functions tackled this way (case by case is popular). Good luck

[astral vine]

To do so you just have to do standard Fourier Analysis

To derive it by hand only from equation on the real variable side is very complicated.

Assume you have a solution on the whole space

(without BC)

Use Fourier Transform on the PDE on the real variable

In space

This leads to an ODE with Fourier multiplier