advanced-pdes

[tall jolt]

clearly not any text on spectral theory in qm discusses this. Yes the hydrogen atom is simple, but I wanted to understand how the business with SO(3) or SO(4) representations generalizes. Maybe the emphasis on the group of the hamiltonian is wrong?

[tall jolt]

maybe an instance of this should be that on a compact Riemannian manifold M, the spectral theory of the Laplacian should be related to the representation theory of Isom(M). But I'm not sure what the right theorem should be right now

actually no ig, because isom(M) can be very small. Maybe this sort of thing will only work for homogeneous spaces (like S^2)

[ripe aurora]

Hey guys, have there been any “attempts” to make something like $\partial y=f_{x_i} (x_1,…,x_i,…,x_n)\partial x_i$ made rigorous?

[untold delta]

Integro-Differo-Topo-Algebro-Cat

[buoyant pike]

Differentials in differential geometry I think is probably what you're looking for

[tall jolt]

tbh I haven't seen representations of this group really discussed anywhere. Most of the time, there are operators that are symmetries already at the classical level that should form a Lie algebra and then you quantize them. This is what happens for the hydrogen atom and gives an so(4) action. But of course there's no general quantization scheme

[ripe aurora]

No, I mean literally have “that”, as it’s written.

[ripe aurora]

I mean literally write the partial derivative by itself.

[waxen bobcat]

That's just notation

[cloud thistle]

it's just not the right way to think about it imo, you should instead think that d(y1, ..., yn) = Df(x1, ..., xi) d(x1, ..., xi)

it gives a more complete picture of how input changes relate to output changes

[exotic lava]

is this supposed to be Einstein summation notation

[sonic jacinth]

maybe a book on non-commutative harmonic analysis if you're interested in how this works for L^2 of some homogeneous space. The thing is that when your space is non-compact, L^2 is now a direct integral over the space of unitary representations of G. this is essentially the non-commutative Fourier transform. I don't know about the general properties of the group U that you descibed though. I think you have to put some strong properties on the operators for this to give you something small enough to be meaningful

[quick pagoda]

For the Hamiltonian part, microlocal stuff is kinda doing this for R^n things

Vaguely

[tall jolt]

yes, but what about the spectral theory of the laplacian

btw do you have a nice reference for this noncommutative fourier transform theorem (in the noncompact case)? Deitmar references Dixmier which is in French

wait maybe this sort of thing is better known in examples like SL2(R)

uhm but I'm not sure when we should expect an intimate relationship between spectral theory and representation theory. That probably happens in like symmetric spaces of finite volume (like H/Gamma where H is the upper half plane and Gamma a fuchsian group of the first kind, possibly with cusps). I have seen the spectral side, but I'm not exactly sure how it relates to the representation theory. I'm sure this is well written somewhere

[sonic jacinth]

On a Lie group G, if you choose a right-invariant metric, then your Laplacian will commute with right translations. Then the point is that you can study the Laplacian on each irrep of G x G separately (sorry, I forgot the relevant rep theory statement here, but I'll try to remember), and then further break it up in terms of right irreps, and it'll act as a scalar on each right irrep of G. On a homogenous space G/H, the situation is different, you still are able to study the action using this decomposition, but now you will stick to looking at left irreps of G, and since left multiplication doesn't commute with the Laplacian, it won't be a scalar on each irrep. sorry, I hope this isn't too confusing, I just woke up

[sonic jacinth]

I learned this from Taylor's book on non-commutative harmonic analysis

[tall jolt]

oh didn't know that book existed, cool, thanks

[sonic jacinth]

so the point I forgot was that right invariant vector fields generate left translation, and this will essentially let you bash out that your Laplacian in fact preserves the decomposition of L^2 into left irreps. From there you can look at the action of the Laplacian on each left irrep separately, if you can figure out nice expressions for these, the spectrum will be much easier to compute

you should compare this to the Fourier transform of the Laplacian on R, it's the same idea

[sonic jacinth]

always. whenever you have symmetry, it will simplify things.
The difference between homogenous spaces with finite volume and general homogenous spaces is that direct sums are replaced with direct integrals. Think like the Fourier transform on R/Z vs on R.

These are a little harder to work with, but the same ideas still work

[sonic jacinth]

I think Taylor works some of this out in his book but believe he doesn't give a complete proof

[tall jolt]

shouldn't there be integrals anyway in the finite volume noncompact case? Because there's nondiscrete spectrum (coming from the Eisenstein series I think). Maybe I'm miswording

[sonic jacinth]

right, I might be mixing things up slightly. In the compact case, things certainly simplify to direct sums.

a priori, there's two ways you can pick up indiscrete spectrum. The first is from a direct integral over irreps, and the second is if the operator on the irreps has an indiscrete spectrum

for non-compact groups, your unitary irreps will generally be infinite dimensional. I don't know of an explicit example where the spectrum on an irrep is not discrete, but that could just be because I know very few examples

[fierce forum]

I think on a compact Lie group G with bi-invariant metric, the Laplacian is the Casimir operator of the bi-regular G×G representation, and Peter-Weyl gives $L^2(G) \cong \widehat{\bigoplus}{\pi \in \widehat{G}} V\pi \otimes V_\pi^*$, with the Laplacian acting as scalar $c(\pi) = -\lambda(\pi)$ on each block. On a compact homogeneous space G/H, the G-invariant Laplacian commutes with the left G-action by definition, so it acts as a scalar on each irreducible G-subspace of $L^2(G/H)$ by Schur's lemma. When (G,H) is a Gelfand pair, each irrep appears at most once and the scalar completely determines the representation

[untold delta]

Beluba

[fierce forum]

On G the laplacian is the quadratic casimir coming from the chosen bi-invariant metric, so the exact scalar and sign depend on normalization, and on G/H it acts by a scalar on each irreducible G-isotypic summand because it lies in the commutant of the left G-action. in the gelfand pair case the spectrum is multiplicity free, but the eigenvalue determines the representation only if different spherical irreps do not share the same casimir eigenvalue, which can fail in general if I recall right away

[sonic jacinth]

no, it doesn't quite work. to define a nice Laplacian on G/H, you need to start with a right invariant Laplacian on G, since it needs to commute with the action of H. but the action of G on G/H is on the left

On compact groups, things are much simpler because you have bi-invariant metrics

[tall jolt]

but in the compact case G/H, with G noncompact, what is the representation theory side

[bright ermine]

I've been learning about non positive-integer Sobolev spaces (things such as H^{-1/2}), defined through the Fourier transform (or Fourier series, when working on tori. In what context are these used? Their definition feels a little bit convoluted

[fierce forum]

You'll still have the decomposition of L^2(G/H) as a discrete direct sum with multiplicities as far as I can tell

[fierce forum]

If I consider the compact riemannian homogeneous case the laplace-beltrami operator is G-invariant and can be realized from a suitable G-invariant metric. It might be satisfied

At this point it matters more if H is compact, and previously we could say something on SL(2,R)/SO(2)

[median forum]

They are a pretty nice description (good functional analysis properties) of some amount of regularity - H^s is -s derivatives away from L^2 etc, in many contexts it is natural to have to consider non integer spaces because you want a sharp result or are interpolating. Also see Schwartz kernel

[tall jolt]

tbh many proofs pass through the fourier transform, so it shouldn't be a strange notion

and like there's many instances of theorems of the form: if s>=n/2 (say) then H^s has this property. n/2 need not be an integer

[median forum]

Yes non-integer spaces exist for W^{s,p}, p\neq 2 as well, and also stuff like Besov. Interesting to note that proving equivalence of fourier definition and "norms weak derivative" definition of H^s spaces requires some nontrivial machinery

[fierce forum]

Given your H^{-1/2} case I find this cool excersie for the invscid SQG equation quite resonant.

Prove that for any $\theta_0 \in L^2$ of zero mean, there exists a global-in-time $L^\infty_t L^2$ weak solution to the Cauchy problem of the SQG Equation.

[untold delta]

Beluba

[fierce forum]

This is a different excersie but it quite demonstrates that SQG system of equation we're talking about in the other problem stated earlier

[fierce forum]

For such proof a special structure of the nonlinear term may be invoked. You may wish to first prove that smooth solutions of the (2.55) equation conserve the quantity $|\theta(t, \cdot)|_{H^{-1/2}}$. One then notes that weak solutions may be defined for any $\theta \in L^\infty_t H^{-1/2}$, by using the $H^{1/2} - H^{-1/2}$ duality pairing.

[untold delta]

Beluba

[fierce forum]

It was originally proved by Resnick for the T^2 case

[buoyant pike]

Oh the S in SQG is for surface?

What an interesting equation to call something quasigeostrophic....

[sonic jacinth]

if G is compact then G/H has a G-invariant metric, but this isn't true in general for non-compact groups. It is true if H is compact, because you can average your left invariant metric on G by the action of H on the right, and then quotient by H.

But it's generally not true if your quotient G/H is compact. For example, there is no SL(2) invariant metric on RP^1

When G is compact, things simplify considerably, but this simplification is mainly because your irreducible unitary representations are finite dimensional, not because of the G-invariant metric.

When G is non-compact, it is still useful to consider how L^2(G/H) decomposes as a direct integral of irreps. The reason is that the Laplace-Beltrami operator is an element of some "universal enveloping algebra", and a unitary representation of G induces a representation of this algebra. So you can compute how the Laplacian acts on each irrep that appears in the direct integral

[fierce forum]

It's funny how the QGPV equation (Quasigeostrophic Potential Vorticity) is more well behaved

[buoyant pike]

What is the QGPV equation

[quick pagoda]

I don’t know why you’d ever want to use a definition other than the Bessel potential for H^s,p fr but the W^s,p haunt me

Mostly because of skill issues though

[quick pagoda]

“Can compute” in a general sense, not necessarily in a way enough to be useful always though 🥀

[verbal nebula]

Everytime, I think it's QSG

[fierce forum]

It's just the tendency equation written or derived in conservative form to describe this Rossby waves propagation in atmospheric things

[untold delta]

Beluba

Beluba

Beluba

[fierce forum]

This is the tendency equation. The second term on RHS can be expanded

[untold delta]

Beluba

[fierce forum]

This is just the $p$-derivative of $f_0 \mathbf{V}_{\mathbf{g}} = \mathbf{k} \times \nabla \Phi$

[untold delta]

Beluba

[fierce forum]

I don't see the point of deriving it here further. But q is completely determined once the 3- dim distribution of geopotential phi is given.

https://arxiv.org/abs/1806.06113

Some people just call the general case "multi-layer quasi-geostrophic" Equations.

[buoyant pike]

Ok so if you have vortex stretching then this is a 3d equation set?

Tbh this feels a bit odd to me

A multilayer vorticity equation

Wait but if you have geopotential, that has to modify the potential vorticity

[fierce forum]

I don't think it's strictly 3d in a sense. each layer is still 2d horizontally, maybe quasi-3d (forgive the terminology) in the sense of stacked 2d layers with vertical coupling, but not fully 3d; the stretching term should let qgpv capture baroclinic effects in a reduced model

It feels so odd

[buoyant pike]

But the geopotential thickness has to change to balance changes in the velocity to the thermal wind relation

So there has to be vertical motion, no?

[sonic jacinth]

yep. to actually extract some information you need to compute what your Laplacian is in the universal enveloping algebra of Lie(G) (the easy part), and then find simple models for the irreps of G (the hard part). plus these also need to be useful

[fierce forum]

Yes there has to be vertical motion even if we can neglect its smaller effect comparatively to the horizontal one so we don't generally violate this balance change. There's this whole Omega Equation in the QG theory that deals with the vertical velocity calculation.

[quick pagoda]

(the hard part) kills me

[buoyant pike]

Better to just use (multi layer) shallow water

[fierce forum]

The numerical versions of QG and RSW models generally have completely different implementations too. There's a certain paper I've seen trying to formulate the former into the latter using projection to acquire a slightly similar implementation. So I generally agree

[digital ibex]

I know for some perturbation function $\Theta (x)$ it must be zero on the boundaries ($\Theta\vert_{x=0,L}= 0$), but is there any restrictions on its derivative(s)?

[untold delta]

KySquared

[median forum]

What context are you working in?

[digital ibex]

I have a coupled system with an initial temperature profile of $T(z) = 1-z$ on $z\in [0,1]$
Im then adding the perturbative term but I just want to think about all conditions $\Theta$ must satisfy

[untold delta]

KySquared

[median forum]

I'm afraid i still don't understand what \Theta is supposed to be>? Are you doing a variational calculation?

[torn dragon]

Hey, does anybody know where I can read about Sobolev space extension theorem? During my sobolev spaces class we only covered the case of C^l class domain, but the lecturer mentioned that it's very tight restriction and that in reality one only need the domain to be Lipschitz. I'd like to read about it a bit more

[fierce forum]

Leoni, Adams & Fournier

[median forum]

I think the Lipschitz extension theorem is in Stein’s singular integrals book

[torn dragon]

guys, thank you very much, I'll look into that

[quick pagoda]

It’s also probably in Evans-Gariepy

[molten viper]

is there always a spherical harmonic H so that it is orthoganol to f1,...,fn but not g? Here the functions are all L2 on the sphere

[hidden coral]

what are f1, ..., fn and g?

[molten viper]

L2 functions on the sphere

can just assume all are linearly independant

[heady silo]

no, and the spherical harmonics aren't important here because they are an L^2 basis for the sphere so so what youre asking is just reducible to a countable infinite dimensional hilbert space and the e_i as your orthonormal basis, there are vectors f that aren't orthogonal to any e_i

[molten viper]

thats pretty much what i figured, basically any function that has no zero fourier coefficients should be a counter example right?

[heady silo]

yep

[molten viper]

ty, but also rip 🙁

[heady silo]

☠️

[solemn swan]

Never would I expect to see universal enveloping algebra in pde wth

[sonic jacinth]

I mean, this is the rep theory side of harmonic analysis, so it makes sense

[solemn swan]

oh

I have no idea what’s going on there lmao

But with that context ig 😭

[prisma pelican]

isn't this a bit weird as a definition of the adjoint? (L being the second order eliptic operator)

coz our base space is H_0^1, not L^2

so $\langle v, Lu\rangle$ does not equal $\int_U v Lu$ but rather [\int_U v_{x_i} (Lu)_{x_i} + v Lu]

[untold delta]

DarQ!

[median forum]

No that’s not the H^1_0 inner product

That also needs like 2 derivatives to work

[prisma pelican]

yea, I'm very confused

[median forum]

Adjoins is always defined through bare integration though, not the specific inner product

[prisma pelican]

bruh

[median forum]

? H^1_0 just has the gradients

But this doesn’t matter

It’s not how you define the adjoint anyway

You are looking at the pairing of a distribution with Lu, not an inner product

[prisma pelican]

yea, I mean, if Lu is in H^1_0 then the gradient exists

[median forum]

Underscore zero doing the work

[prisma pelican]

I checked, and H_0^1 has the same inner product as H^1

in brezis at any rate

[median forum]

There is a poincare inequality so it doesn't matter

BUT- this is not the point

[median forum]

this is

[prisma pelican]

I don't know what that means, however

evans does not touch distributions at all

[median forum]

Notice that Evans is using \langle, \rangle here instead of ()

[prisma pelican]

right

[median forum]

This means that you are not pairing two functions in H^1_0, but pairing with something in the dual space

When you pair a Sobolev function against someething in its dual you always just do this by integrating - without taking any derivatives

Does this make sense?

[prisma pelican]

ok, now I'm very confused

maybe it's lowk just a typo?

[median forum]

It's not

[prisma pelican]

ok, no, I'm so silly, evans doesn't use the < > coupling at all, I did

anywho, I think it'll make more sense why we are using this choice of the definition of the adjoint once I see the subsequent proof

[median forum]

It should make sense now

[prisma pelican]

I dunno, I don't see it, obviously by using ( , ) we're considering H_0^1 as a subspace of L^2 but I don't know why

[median forum]

Like if you just pair $\langle u, Lv\rangle$ this is equal to $\langle L^* v, u\rangle$ if you use the right defintiions

[untold delta]

zeke

[median forum]

You don't want to use (,)

[prisma pelican]

the ( , ) coupling is the definition which gives us (v, Lu) =(L*v, u), though

(u,v) being \int u v

[median forum]

Where is it defined this way?

[prisma pelican]

.

[median forum]

That's not what this is saying

[median forum]

The RHS is the forcing

It's not the derivatives

[prisma pelican]

where ( , ) denotes the inner product in L^2(U)?

[median forum]

What does this have to do with the adjoint

This is just a different statement

[prisma pelican]

.

evans is distinguishing between <, > and (, )

[median forum]

You are just wrong though

[prisma pelican]

bruh

[median forum]

You are pointing to a completely different statement

[prisma pelican]

what, why would (, ) two different things in the same chapter? 😭

are you suggesting (, ) and < ,> switched definitions mid chapter, what

[median forum]

Where is the adjoint defined using (,)?

[prisma pelican]

it's not defined using (, ), I'm saying it satisfies (v, Lu) = (L*v, u)

where L* is this

[median forum]

I think this is "wrong" - it is correct in the end, but it misses the point of what is happening

[prisma pelican]

and Lu is this

[median forum]

Lu is in (H^1_0)^* so we should be using the dual pairing

However the pairing just happens to be integrating against each other

[glossy heart]

Huh this perator is weird

are those partial derivatives?

$Lu = \sum_{i,j} \partial_j (a^{ij} \partial_i u) + b \cdot \nabla u + c u$

[untold delta]

TC159

[glossy heart]

I wonder how to simplify the sum, I assume it can be done

$Lu = \nabla \cdot (A\nabla u) + b \cdot \nabla u + cu?$

[untold delta]

TC159

[glossy heart]

Perhaps its the transpose, but oh well I don't wanna write it down

looks like it's this

[glossy heart]

Oh god this notation is horrible

[quick pagoda]

It’s that up to a different A fr

[glossy heart]

I fucking hate it when shit is written in terms of sums

like this

also you have Poincarés-inequality, you can just define the inner product in H^1_0 in terms of the gradients

[quick pagoda]

It’s handy sometimes

[glossy heart]

Yeah, but it being used as default is fucked up

some shit is basis / orientation independent but sometimes obfuscated by the sums

[quick pagoda]

$L^u = (-\div A\grad u + b\cdot\grad u + cu)^ = -\div A\grad u - b\cdot \grad u + (c-\div b)u$

[untold delta]

Sharp, Archon of Severing

[glossy heart]

Meanwhile I look at this expression and I'm thinking about "oh shit, what are shit shits like the divergence theorem and shit, I can probably use this to derive a weak form for duality with u"

(I am horrible at remembering the multivariable integration theorems xd)

[quick pagoda]

Real

[glossy heart]

Everything is just integration by parts in some way or just stokes'

[quick pagoda]

Sign errors my behated

The shenanigans one bad boundaries sucks too

[glossy heart]

You assume the boundary is smooth until you write the proof, that everything is good, and then you see what you actually need

Anyway

[quick pagoda]

Doesn’t work if the point is to get past that easy regularity case

[glossy heart]

fuck H^1_0

Why work in L^2 when full L^p generality

[quick pagoda]

See above complaints about how some embedding norms blow up

[quick pagoda]

Things being false

[glossy heart]

I'm starting to think that you're interested in low regularity boundary conditions

[quick pagoda]

Shocking I know

But yeah there’s terrible things that happen outside of L^2

Like complete failure of some desired well-posedness results

[glossy heart]

Ok well, first of all Lp means p in (1,+infty) with the cases 1 and infty being dealt with separately

[quick pagoda]

I mean there’s some stuff for the wave equation that’s true iff p = 2

[glossy heart]

mfw results in the weak topology, and then you have Linfty and you have to deal with weak star topology

[glossy heart]

which one is the wave equation?

I really have a hard time remembering PDEs by name xd

[quick pagoda]

$\pdv{^2}{t^2}-\Delta$

[untold delta]

Sharp, Archon of Severing

[glossy heart]

I mean yeah the laplacian is kinda L2 based

I'm sure you have some results with a version for p-laplacians

or something

[quick pagoda]

Weak star is not that terrible for L^\infty, it’s L^1 that turbo sucks

[quick pagoda]

I believe no due to some exponential weights?

[glossy heart]

I just mean it's annoying to write
"for all 1<= p < +infty this converges in the weak topology. In the case of p = +infty this converges in the weak * topology"

lmao

[quick pagoda]

And nonlinear issues

[glossy heart]

Anyway yeah that makes sense

[quick pagoda]

In everything except 1, it’s all weak*

[glossy heart]

Geometric objects, not interacting well with non-geometric objects makes sense

[glossy heart]

They're reflexive so of course weak is weak* 😭

[quick pagoda]

1 is always the problem child

[glossy heart]

I dunno, C(X) for me is closer to Lp than Linfty

[quick pagoda]

I don’t think it’s too fair to say that L^p is non-geometric

[glossy heart]

Sure, but you can't expect Euclidean operators, like the Laplacian to interact well with non-Euclidean objects

maybe I should've written it more like that

Where everything is meant to be interpreted in an intuitive way, without formal justification

[quick pagoda]

I mean it does work well enough for the laplacian though

Just not with the two t derivatives

[glossy heart]

oh okay

mb

[quick pagoda]

Works fine with one t derivative too iirc

[glossy heart]

My PDEs knowledge is mostly due to one extremely intensive calculus of variations course I took last year

[quick pagoda]

But there’s usual positive/negative t slop

[glossy heart]

We finish the whole book and started going to Brezis and Evans after

[quick pagoda]

Parabolic regularity doesn’t work for hyperbolic etc

[glossy heart]

So like PDEs appeared in the context of like EL, and the professor discussed how the modern approach for uniqueness and existence in PDEs is to associate it with a functional

[glossy heart]

Pretty dope classes, the professor just gave an intuitive understanding of things but we were off to fend off on the book by ourselves

I remember one time I proved a functional was lower unbounded without computing a single value by proving that boundedness contradicts like Poincare or something using a similar in spirit argument to bootstrapping. Mostly because I was lazy to actually evaluate the functional though

[tall jolt]

Does anybody know a reference for this theorem? I could find the original article here https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mmo&paperid=97&option_lang=eng, but it's in Russian

here H_N is the N electrons, Z protons schrodinger operator

[glossy heart]

Is it EL?

[tall jolt]

what? https://www.math.lmu.de/~nam/Ionization-Lieb-Collection.pdf

[glossy heart]

Euler-Lagrange

[tall jolt]

I'm not sure if it's literally euler-lagrange, but there's some variational principle relating (4) and (5), as they say

[glossy heart]

Anything that's basically "modify an operator, so that you have an expression which integrates 0 against test functions" I call EL. That being said I have not read it. I'm not sure what H_N is

[tall jolt]

ok?

[prisma pelican]

yea

[prisma pelican]

I think so, yea

[vagrant merlin]

Hello, I am looking for a name for functions $f: \real\to\real$ with the property
[
f(x) \leq \limsup_{y\uparrow x} f(y) = \inf_{r>0}\sup_{0<x-y\leq r} f(y) \quad \forall x\in \real
]
This is weaker than left lsc. Under some additional more assumptions it allows for the left continuity of $g$, where
[
g(x) = \sup_{y\leq x} f(y)
]
I think, if we assume $\Gamma(x) = {y\geq x,: g(y) = f(y)}$ not being empty at every $x$, then $g$ has to be left continuous. Similar statements hold for $\sup$ replaced with $\inf$, and ${y\leq x}$ replaced with ${y\geq x}$.

[valid mulch]

what are you working in @vagrant merlin

There isn't one universally standard short name, but in analysis literature these are usually described as:

Left upper semicontinuous functions
or
Upper semicontinuous from the left

because the inequality mirrors the usual upper semicontinuity condition but using one-sided limits.

So the most common phrase is:

left upper semicontinuous (left USC)

​

[vagrant merlin]

I am working in stochastic control theory

[valid mulch]

one second i may have something let me work this out ill write it in latex

[vagrant merlin]

The condition is strictly weaker than left LSC
Recall that USC implies
[
\limsup_{y\to x_0} f(y)\leq f(x_0)
]

[valid mulch]

Your image is about upper semicontinuity (USC), while earlier discussion was about delta functions collapsing variables in the Klein–Gordon commutator. The logical mechanism is similar though: a limit condition or constraint removes extra terms.

[untold delta]

WildNapkins

[valid mulch]

does this help?

i dont understand these reactions yall are leaving lol

[fierce forum]

Stop using AI

[median forum]

you posted what seems to be AI slop completely unrelated to the discussion

What does this have to do with Klein Gordon

[vagrant merlin]

No problem at all — thank you for the clarification. I appreciate you taking the time to correct that. Interpreting it in terms of lower semicontinuity is helpful, and I agree that the condition under discussion is strictly weaker. It’s completely understandable how that mix-up could happen across multiple threads.

[untold delta]

WildNapkins

WildNapkins

[vagrant merlin]

<@&268886789983436800> wildnapkins chagpt

[cinder oasis]

[cinder oasis]

https://tenor.com/view/eminem-crying-meme-gif-3627991839743427357

[prisma pelican]

ok, why is the very first proof of regularity I ever see which uses energy methods is like

5 pages long

chat is it joever?

[prisma pelican]

take u\in H^1(U), and suppose zeta has this condition:

take $v=-D^{-h}_k(\zeta^2 D^h_k u)$

[untold delta]

DarQ!

[buoyant pike]

What is sigma?

[prisma pelican]

the first equality here comes from the fact that \zeta^2 D_k^h u is in H^1 so difference quotients are bounded

[prisma pelican]

zeta, sorry

where does the 2nd inequality come from?

is it poincaré or smth?

this is the difference quotient if that's necessary

I'm rather sure these are the only assumptions you need on u and v

but feel free to ping in case more context is needed

(this is the 5 page proof in question btw 😭)

[median forum]

product rule, first term differentiates the cutoff

You get to go to W bc of the support of \zeta probably

[prisma pelican]

oh right, I'm fucking high

ugh

this whole proof makes no sense to me

each step makes sense ig but I have no idea how we came to find these steps

and why they prove the thing we want ot prove

[median forum]

Does the set up make sense at least? \

[prisma pelican]

yea, you're just trying to prove the quotient differences are bounded in norm

[median forum]

Yeah I mean the idea is pretty typical - we want to get regularity, but we will start we interior regularity so we need to take a cutoff, in addition, since we only have a weak solution we need to use some sort of mollification of the derivatives - here a difference quotient, get the estimate uniformly in terms of the mollifying parameter and then take a limit

IMO difference quotients are annoying but skill issue maybe

[stone fiber]

[this regards elliptic equation theory]
does anyone know how i could potentially convert a problem like[ \begin{dcases}-u''+p(x)u+x\int_0^1 y u dy=f & \text{almost everywhere on } (0,1)\
u(0)=ku(1), u'(1)=ku'(0) & \end{dcases},]
where $f\in L^2(0,1), p\in C([0,1]), k\neq 1$, and apriori we know $u$ is in $H^1(0,1)=W^{2,1}(0,1)$ (sobolev space), for which we've proved it satisfies the weak form of the above problem, such that the left side is a linear differential operator? We have a theorem that tells us if the first line in the above is a second-order differential operator, then not only does the weak solution (the one concerning the associated bilinear symmetric bounded coercive form, see Lax-Milgram) satisfy the strong problem, but also is more regular, $u\in H^2=W^{2,2}$. I just don't see how i can apply this, since the above functional is not immediately obvious to have the form[ L=-\Delta\underline{\phantom{m}}+\langle b,\nabla\underline{\phantom{m}}\rangle+c\underline{\phantom{m}}]for some function $c$ and mapping $b$.

[untold delta]

🇵🇸μ🔆

[buoyant pike]

Why do you think this is elliptic

[stone fiber]

i mean it basically is, right.
like from what i was supposed to do (namely apply the L-M thm) all is very suggestive to somehow manipulate this problem to turn it into the 2O diff operator form

keep in mind analysis is not my field i only do this cuz i have to.

[buoyant pike]

Is this a homework problem

[buoyant pike]

Oh wait

Have you tried integrating by parts on the integral

[stone fiber]

idk how that helps[-u''+pu+x\int_0^1yudy=-\Delta u+pu+x(\tfrac{1}{2}u(1)-\int_0^1u'\frac{y^2}{2}dy)]
i just have a different integral form

[untold delta]

🇵🇸μ🔆

[median forum]

You can do the integral

[stone fiber]

am i just stupid, what am i missing

u is a solution of a form i don't necessarily have

[median forum]

My bad I’m wrong

[stone fiber]

so like

i know that i have a solution to the weak problem)

i can get a solution by using a bilinear form associated to this

but i just don't know about the better regularity since this functional is not differential.

[median forum]

I mean have you tried to just use the equation to bound the L^2 norm of u’’

What

[stone fiber]

sry

ik what you mean now. to prove u is in H^2

i haven't had much sleep, apologies

[prisma pelican]

let $\zeta$ be a cut off function on $V\subset\subset W\subset\subset U$ and $u\in H^1(U)$. And suppose $a^{ij}\in C^1(U)$.

[untold delta]

DarQ!

[prisma pelican]

let A_2 be this

how do you arrive at this inequality?

[stiff veldt]

is that supposed to be A_2 or A_1

[prisma pelican]

A_2, sorry

most of this makes sense except for why \zeta_x_j disappeared

coz as far as I can tell we have no other assumptions on the cut off function

[stiff veldt]

yeah because im pretty sure you just use smoothness to deduce all the a_{ij} factors are uniformly bounded and whatnot

so most of the junk just gets absorbed into C

[prisma pelican]

yes

also, \zeta^2<\zeta so that's also figured out

I'm able to justify everything except \zeta_x_j

[stiff veldt]

is there any reason you cant just bound \zeta_{x_j} in the exact same way

[prisma pelican]

you could be then the C would depend on our choice of \zeta

we want the C to only depend on U, V, and the coefficients

and \zeta_x_j could be arbitrarily large so that's not very useful

[stiff veldt]

the choice of \zeta itself really depends on U, V though no

[prisma pelican]

it depends on the choice of W

if W is super tight then \zeta_x_j is large

[stiff veldt]

im aware of this

but my point is that all your choices ultimately boil down to whatever U, V happen to be

[prisma pelican]

hmm

[stiff veldt]

maybe its a wording issue? idk

i feel like its just the language evans uses

[prisma pelican]

I'm trying to make sense of C only depending on U, V, and the coefficients ig

I think you're right

maybe C does depend on our choice of W but there is a smallest C which gives us the inequality we desire

and that does not depend on W

[stiff veldt]

i remember reading this exact proof and i think i wondered something similar myself, but in a sense W and \zeta both depend on U, V so idk if it really matters

[prisma pelican]

this proof is lowk killing me lmao

[stiff veldt]

for example you could take W to be "halfway" in between U and V, then \zeta only depends on those two

[prisma pelican]

I hate regularity proofs

[stiff veldt]

yeah i slogged through that regularity proof and was like "im good" after seeing the next one ngl

i need to get back to it at some point though

i think after my sieve theory experiences i will be able to tolerate what im looking at a bit more

[stiff veldt]

anyway maybe im wrong about this but its the conclusion i came to alone, if someone else has another opinion im willing to hear it

[prisma pelican]

LMFAO

yea, like, interior regularity looks disgusting enough

boundary regularity sounds

[stiff veldt]

yeah i think i skimmed it and was like "wow this looks awful"

maybe i will just have to read all the regularity proofs in evans to get a handle of what they are actually doing

like none of the steps individually is unreasonable but you cant help but wonder why this is the method being used

[prisma pelican]

yea, exactly

in this particular proof ig the idea is to just bound the norm of the quotient difference

and then use this characterization

but like

how you come to bound it is distressing

even if someone gave me the choice of v I need to use and the inequality I need to prove it would still take me weeks to come up with a proof like this

[stiff veldt]

yeah now that i look at it (2) is really the key

thats the actual regularity thing which allows you to get an extra derivative

the computations well

i know there is a lot of integration by parts and other nonsense but i will have to digest why this is a good way to proceed

i should probably just read the other regularity proofs and look for similarities

[prisma pelican]

that's actually a good idea

[prisma pelican]

I want to verify whether using this is like

the only idea to get an extra derivative

or more generally proving u is in H^2 instead of proving it's in C^2

and then using some sobolev embedding

[stiff veldt]

a fair question, unfortunately i have no clue

[prisma pelican]

thanks for the sanity check btw!

[stiff veldt]

no problem, we are all suffering together

[prisma pelican]

just to make sure I understand what theorem 4 is really saying

so theorem 1 tells us u has a 2nd weak derivative on all of U

but we can't conclude u is in H^2 since we're not sure its norm is finite

and that's exactly what theorem 4 says

it says the C in theorem 1 actually can be chosen to only depend on U and u is bounded in H^2

is that correct? like, is that the actual significance of theorem 4 or am I missing something?

[fierce forum]

Are you interpreting it as all of U upto the boundary?

[prisma pelican]

well, u is in H^1 so derivatives on the boundary don't make any sense

also, U is open so it doesn't include its boudnary

[stiff veldt]

yeah more or less

if V approximates U in the first result then you could have blowup of the constant C so you cant really conclude anything nice

[fierce forum]

This dependence is right to notice yes

[fierce forum]

I know. I just meant the loc regularity part as on how you interpret it even if u isn't H^1 (it appears non relevant or trivial to you but I wanted to know your interpretation)

[untold delta]

Beluba

[fierce forum]

This implies that, in the generic case, solutions of (ES) can not remain bounded in any high Sobolev space, which embeds into W^{1,\infty} and stating the proof makes the argument clearer.

Almost all of the cases I'm intrested in are answered negatively in low regularity but affirmatively in high regularity. Like those nonlinear invscid damping results that has certain unknown regularity dynamics still

The literature reaches it from low regularity first as a treatment. That's why getting the low regularity bits initially is very reasonable @quick pagoda

It is initially reasonable to read the work of Mouhot and Villani concerning the the Vlasov-Poisson system such as this paper

https://arxiv.org/abs/0904.2760

[quick pagoda]

Makes sense

[fierce forum]

Then one could read the Bedrossian and Masmoudi paper for the invscid damping for infinite channel, and could then follow it with the Ionescu/Jia paper with a boundary/finite channel. The negative low regularity results is primarily in the paper by Lin and Zeng

[quick pagoda]

Low regularity has false things, high regularity only happens locally…..

[fierce forum]

Yeah it gives pretty false things

[fierce forum]

You would find a survey of this and other advanced results in the fluid text of Vlad & Bedrossian which is quite nice for beginning stuff

If you see a book named Galdi don't open it until you've been prepared a bit

[quick pagoda]

Noted

[quick pagoda]

Noted noted

Right, there’s a lot of problems that seem to occur

[fierce forum]

I haven't mentioned Majda and Bertozzi or Temam etc

[quick pagoda]

Things blowing up is bad, but if there were something handy like some particular weighted/orlicz-y global in time solution stuff that would be also curious

[fierce forum]

I have thought in this direction a while ago but it would just require to explicitly resolve the boundary layer which is kind of weak

[quick pagoda]

Yeah understandable

Anything like upper bounds on how the norm can blow up at infinity, when possible, is always fun

[fierce forum]

This is indeed fun. You'd have a nice opportunity raising and thinking of solutions around the next case at the end of this paper https://arxiv.org/abs/1506.04010

[quick pagoda]

Curious but I doubt I’ll get much materialized from it, something to look into though

[velvet pier]

ig just u\in H^1(V) for V \subset\subset U

[prisma pelican]

are all papers this big?

the average paper being like a hundred page in the least? 😭

[median forum]

Welcome to the wonderful world of PDEs /s

[fierce forum]

Not necessarily lol

There are many short papers but that doesn't mean less dense

[prisma pelican]

I see

[verbal nebula]

Usually papers this long have a lot of the theory/background developed within it

Shorter papers usually just assume familiarity with the recent literature it's citing

Stan Palasek, Mimi Dai, and Cheskidov have a paper of this instantaneous blow up of solutions to Navier-Stokes equations from the right hand-side

It's like 50 pages, with 20 pages just being the intro. The actual arguments don't last too long, but are incredibly dense

[fierce forum]

I remember this paper it came in my mind when I was talking with sharp about the BMO^-1 space (in the work by Koch and Tataru which was a quite short paper communicated by Charles Fefferman)

[verbal nebula]

Yeah, it's a decent paper. But hard to follow

[quick pagoda]

Every time

[quick pagoda]

[fierce forum]

I remember that it was being discussed at your Fluids Seminar when you were supposed to be speaking, and if I recall your advisor and a postdoc were getting stressed just by the title or the very beginning of the paper? It seemed strange to all

Yeah this is what I said in that discussion back then. I've gone through it in depth more but it's still quite dense

[verbal nebula]

Yeah! I was presenting on Roger Temam's paper proving existence and uniqueness to the 2D euler equations

I've now read the Instantaneous blow-up a couple times, each pass going through more detail with a professor that specializes on regularity issues

It's a very opaque paper, that seems to assume familiarity with many strange looking decompositions; and the geometric set up is not clear at all

[stiff veldt]

<@&268886789983436800>

[digital ibex]

Does anyone know of the derivation for the Allen-Cahn or Cahn-Hilliard equations? I can’t seem to find anything online about it

[buoyant pike]

Have you read the 1958 Cahn Hilliard paper

[digital ibex]

No
Could you provide the paper’s title?

Does that paper derive the A-C equation?
Ik the C-H equation is an extension of the former so ideally I’d like to work my way up

[buoyant pike]

C-H came first

https://pubs.aip.org/aip/jcp/article-abstract/28/2/258/74794/Free-Energy-of-a-Nonuniform-System-I-Interfacial

1958

[digital ibex]

Really? I didn’t know that

Ig in that case, A-C is a specific case of C-H, right?

[buoyant pike]

https://www.sciencedirect.com/science/article/abs/pii/0001616072900375

And this should be the A-C paper

[prisma pelican]

the proof of this lemma is straight up delirium 😭

the lemma itself is lowk random af

[stiff veldt]

i think the lemma itself is reasonable but the proof technique is a bit odd

[quick pagoda]

What is c here

But yeah, on a contrapositive pov, 0 normal derivative at x_0 -> u(x_0) cannot be greater than everything in U

So it's odd conditions but, put another way, it's a way to control u based on boundary normal derivative values, which is a strong way to say things cannot vanish everywhere while u does not (if Lu has some comparison principles, nice properties, etc whatevers)

It's just phrased in a curious way to separate the u conditions from the normal derivative conditions

[buoyant pike]

c is the constant in the elliptic operator L I think

[quick pagoda]

I see I see, makes sense? I guess? Seems odd for that to be zero

Or maybe I'm imagining a different one than intended

[graceful socket]

hi, a basic question about McKean-Vlasov equations

to work in this setting, we must presume we're working 1) with large populations of whatever (people, neurons, particles, asks and bids in a stock exchange etc) and 2) we should presume that as the amout of interacting objects we're describing goes to infinity, the influence of any object towards another object is negligible, so we can treat the whole population using one mean representative, right?

[cloud thistle]

yes exactly, also included in the latter point is exchangeability, swapping the identity of two birds or two particles doesn't change the distribution because every part of the system is "anonymous"

which is why studying the empirical measure is justified

the influence between any particular objects becoming negligible is called "propagation of chaos"

[graceful socket]

thanks

ive been trying to understand what the difference between HJB+FPK coupled equations machinery and McKean-Vlasov equations is

it seems that McKean-Vlasov describe this mean representatives behaviour whereas FPK describe the whole population distribution denseness here and there, and HJB is describing the most optimal way for an object to reach the desired goal

[cloud thistle]

every mckean vlasov process has a FPK equation

adding on HJB turns it into a mean-field game system, it encodes the assumption that every agent tries to rationally maximize their utility (with perfect foresight)

[graceful socket]

so HJB part implies that every agent wants to reach some goal?

and shows how its done if it chooses the optimal path

[cloud thistle]

yea exactly, if you know about dynamic programming from an algorithms class, HJB is just dynamic programming, trying to optimize some goal function

[graceful socket]

nice

and if i may

how does it interact with the mean field control tools?

hjb is used there a lot i believe?

[cloud thistle]

the optimization problem each agent is trying to solve is an optimal control problem

each agent finds an optimal policy (whatever actions the agent can take over time) in the sense of maximizing their goal function

mean-field game theory is just combining optimal control with mckean–vlasov diffusion

[graceful socket]

i mean ive seen mean field control being described as a holistic area of research, sorry if i brought a non existent thing into the conversation

there was a mention of MFG and MFC being distinct areas of math

somewhere

so i thought that was a thing of its own

[cloud thistle]

yea they're both things

here's a paper on mean-field optimal control: https://www.numdam.org/item/COCV_2014__20_4_1123_0/

we address the situation where the individuals are actually influenced also by an external policy maker

[graceful socket]

hmmm thats what control stands for here

external policy maker

[cloud thistle]

I don't know too much about mean-field control, but there seems to be a nice introduction to MFGs and MFC here https://www.pnas.org/doi/10.1073/pnas.1922204117

Mathematically, MFC models are similar to potential MFGs. The key difference in MFC is that a central planner devises an optimal strategy

[graceful socket]

ill look into both

btw ive heard that Carmona & Delarue 2 vol set on MFG is the most comprehensive treatment of the subject as of now

do you think its a good point to start? in the prologue they state they were trying to make it self-contained and as pedagogical as possible

[graceful socket]

thats actually pretty sick, i was just thinking of how we can solve it using ML tools

[cloud thistle]

I've read a little of it, it is pretty great but long

[graceful socket]

all of the papers on MFG in general seem to be pretty recent too btw

[cloud thistle]

yea definitely

if you go to my thesis, I link a good number of sources at the end of each section

those are the best ones I found, for each topic

[graceful socket]

ive seen it yeah

probably where i found this 2 volumes reference lol

[cloud thistle]

hahaha

I like it, it does feel very motivated and fairly self-contained

[cloud thistle]

the whole field started only in the 2010s

[graceful socket]

yeah

also ive just looked into your thesis and you mention Pierre Cardaliaguet there whose lecture on MFG ive found just a few hours ago lol

the field seems to be quite small huh

[cloud thistle]

Yup he's one of the original inventors

He has the influential paper about the master equation and systems with common noise

[graceful socket]

master equations look scary

[graceful socket]

i found out that Lions is considered one of the founders of the theory

[cloud thistle]

yeah lasry and lions were the starters

carmona & delarue, and cardaliaguet are the other big names

[graceful socket]

in Carmona & Delarue they mention his L-differential calculus and Lions derivatives in the very beginning

[cloud thistle]

carmona & delarue work more on the stochastic control side I think

[graceful socket]

yeah they mention they work in the probabilistic setting

[cloud thistle]

yea

[graceful socket]

(idk which other setting there's though??)

like looking at MFG through PDE lens solely or something

[cloud thistle]

yeah

something I enjoyed about mean-field games is that it unites a lot of distinct areas of math together

[graceful socket]

and a huge ton of applications lolol

[cloud thistle]

I was mostly using it as an excuse to learn more math rather than the specific theory or modeling aspects

haha

I'm not convinced by the applications, I think there hasn't been enough time passed to assess how useful they've been

but I gave a few in my thesis anyways for fun

[graceful socket]

the idea of looking how large amounts of agents interact with each other and sense the mean feel of the population rather than their immediate neighbours influence seems to me rather plausible and applicable in nature

anyways I've read somewhere that the mathematical framework came from statistical physics initially

so I'll just think it was motivated by nature processes themselves lolol

[graceful socket]

thats partly why when i learned that such an area of math existed i immediately became interested

anything involving different kinds of cool math is great if its not connected to algebra

[cloud thistle]

hahahahaha

[cloud thistle]

yes, mean-field theory came from studies of particles in a gas

david tong has some nice notes on a two-semester course in statistical physics

and mit ocw also has lecture videos on it

[cloud thistle]

the assumption that all agents interact "anonymously" with identical strategies feels very strong, combined with the assumption of perfect rationality

I'd be interested to see how evolutionary game theory fits into the picture

[graceful socket]

isnt anonymous here equivalent to treating any agent interchangeably with another? or what new information does the property of being anonymous bring

and perfect rationality seems to be how cells act (thats the application im most interested in)

[cloud thistle]

Yeah, I'm thinking about for example human social networks: I certainly don't interact with my friends the same way I interact with random strangers

It's unclear to me empirically how much you can reduce e.g. epidemiology on graphs down to these mean-field game systems

[cloud thistle]

I imagine cells mostly interact locally though, right? like it's strange for the action of a cell to depend on the action of cells in state space that might be located very far physically

[graceful socket]

yep they act based on what cells surround them, so distribution of cells in space matters

[graceful socket]

oh i think ive seen it under the name of weighted agents or something, when you entitle some with more influence than others

[cloud thistle]

That being said, I think it's really nice as a normative model so that you can compare the behavior of biological systems to predictions from rational agents and see if it matches up or how it deviates

Normative models are always welcome in biology

[vagrant merlin]

what area are u working in?

[cloud thistle]

I wrote my master's thesis as an introduction to mean-field game theory

[prisma pelican]

the coefficient of u in L

[quick pagoda]

Ah that makes more sense

Yeah ok

[high canopy]

What are tips to non dimensionalising a pde?

in my exams its just annoying because i cant seem to get their answer

[cloud thistle]

Can you give an example of an equation you're struggling to nondimensionalize

[graceful socket]

tensor train decomposition

[high canopy]

viscous burgers equation

[high canopy]

how do i solve this btw?

[valid scarab]

you could use the method of characteristics since its a first order PDE

[prisma pelican]

<@&268886789983436800> clear gpt response

and the account is just advertising their tutoring service

(sorry, I'm on my studying account)

[vestal seal]

Yeah, I'm not particularly convinced you've done these yourself tbh

[valid scarab]

wikipedia is enough, i think. you can find pdfs from stanford or cambridge too, that are very good

[cinder oasis]

i was gonna reply with "gpt slop" but wasnt sure so i didnt

[loud patrol]

I dont understand distributions im not sure how to prove how something is a fundamental solution to a PDE using the definition Let $\Omega \subset \mathbb{R}^n$ be open and let $a_\alpha(x) \in C^\infty(\Omega)$. Consider a linear differential operator $L = \sum_{|\alpha|\leq m} a_\alpha(x)\partial^\alpha$. We say that a distribution $E \in \mathcal{D}(\Omega)$ is a fundamental solution of L in $\Omega$ if $LE = \delta$

[untold delta]

n

[loud patrol]

Okay so lets say you havea pde

is LE "multiplication" of the solution and linear differential operator (which im assuming is the pde we are tryint to show E is a solution to)

im just in general confused how to show that you obtain LE is equal to the dirac delta like if you are using a test function to define LE for example, (E(x,t) is the heat kernel) - like LE is defined as following but why as an integral $-\int_{\mathbb{R}^{n+1}}E(x,t)[\partial_t\varphi(x,t)+\Delta \varphi(x,t) ] \ dx \ dt$

[untold delta]

n

[sonic olive]

It is delta as a distribution in the sense that you need to apply it to a test function

[loud patrol]

ive been looking at the wikipedia page for distributions but idk its more or less helpful

[loud patrol]

yea i guess i get that but how would i prove such thing

[loud patrol]

like show this integral ends up being the dirac delta at the origin

[sonic olive]

Usually it involves some sort of integration by parts and special properties of your kernel

It’s not just something you can do completely abstractly

[loud patrol]

oh

hmm

[stray forum]

Can anyone please help me to solve it, really urgently needed it....

I understand the maximum principle will be used but don't understand how?

[quick pagoda]

What have you tried

[stray forum]

Take some v(x,y)=e^2x cos(2y) then consider like u(x,y)-\epsilon v(x,y). Butdid not work really it is not my idea gpts but i find the error there.

[loud patrol]

can someone check my proofs please? with showing that the heat kernel is a fundamental solution to the heatoperator

using the theory of distrubutions

[quick pagoda]

Don't use GPT, try considering your maximum principle argument

What stops it from just being "apply maximum"

[valid scarab]

maybe you could try v(x,y) = cosh(cx) * cos(cy) and search for a fitting constant c. I guess c = 2 fits the condition from Omega

[prisma pelican]

ok, so huge technical proof incoming

I want to make sure I haven't glossed over any underdevelopped technical conclusion, so (25) is true because, first, D_pL(Du, u_k, x) is a linear functional on (L^p)^n

since it's bounded on G_\epsilon and U has finite measure

and then we're implicitly using (iv) here, right?

this is the only reason why I could explain our choice of F_\epsilon and E_\epsilon

to make D_pL(Du, u_k, x) converge in norm (as a linear functional)

[stray forum]

Urgent help needed a problem please.

[stiff veldt]

yeah i guess that does work, my notes have a different justification

[prisma pelican]

please do say

[prisma pelican]

why is it so urgent?

is your deadline drawing near or smth?

[stray forum]

It was some part of posdible question in exam, and professor asked however possible try to solve this before coming to exam. I tried nothing clicked sent it here.

[stiff veldt]

i had some comment about uniform boundedness but frankly i dont know what the intent of this line of reasoning was anymore

i like your justification better anyways

when i read this proof initially i think my main gripe was with (24)

[prisma pelican]

(24) is just the DCT, right?

wait, no, not precisely, hmm

[stiff veldt]

i remember (25) also gave me a lot of trouble but (24) raised an eyebrow because in principle you dont need uniform convergence of L(Du, u_k, x) to L(Du, u, x)

uniform convergence doesnt necessarily behave well under composition

[prisma pelican]

it doesn't? even if L is continuous?

[stiff veldt]

with uniform continuity its fine, but i think regular old continuity is technically not strong enough

[prisma pelican]

right

[stiff veldt]

i think for this proof its ok because uniform convergence implies (u_k) is uniformly bounded, so if you restrict to some large enough compact set then L(p, z, x) is uniformly continuous

[prisma pelican]

yea, that's what I'm thinking

[stiff veldt]

yeah

just in principle it might go wrong, but here its ok

[prisma pelican]

I mean, you don't even need uniform convergence of L(Du, u_k, x), right?

you just need a.e. convergence

and then, since L(Du, u_k, x) is bounded and U has finite measure then you just use the DCT

now that I think about it, same for D_pL(Du, u_k, x)

you dont' need uniform convergence for convergence in L^p'

[stiff veldt]

yeah i guess DCT is fine

i cant think of any reason it would fail

i think the main point is that (u_k) is uniformly bounded

and then the other coordinates are checked by definition of F_epsilon

[prisma pelican]

well, now I'm confused

if we didn't use uniform convergence in (24) nor in (25) then why did we use egoroff to get a uniformly convergent subsequence?

as far as I can tell, we just need F_\epsilon

[stiff veldt]

uniform convergence of u_k -> u implies (u_k) is uniformly bounded and therefore so is L(Du, u_k, x)

if you didnt know it was uniformly bounded then DCT could fail

[prisma pelican]

ohhhhh

yea, right

that makes sense

I forgor

[stiff veldt]

i got too stuck thinking about uniform convergence => interchange of integrals

but DCT is easier for this

uniform boundedness still matters though

[prisma pelican]

yea

our life is so infinitely easy here coz U has finite measure

[stiff veldt]

yeah if it did not this argument fails terribly lol

[quick pagoda]

What have you tried

[sonic olive]

Try converting the setup into weak form and plugging in a test function that would imply the result

[sonic tapir]

https://math.stackexchange.com/questions/655905/evans-pde-chapter-2-problem-4

I do not understand the line "If uε attained a local maximum at an interior point then Δu≤0."

[stiff veldt]

the laplacian of u is the trace of its hessian, and at any interior local maximum the hessian is negative semidefinite

so the trace and therefore the laplacian is <= 0

[sonic tapir]

thanks!

[meager dune]

Tbf the sentence also is weirdly written

[cloud thistle]

<@&268886789983436800> spamming across many channels

[pulsar forge]

Dont spam your message, use one channel.

[buoyant pike]

Are the messages in the other channels going to be cleaned up?

[pulsar forge]

huh weird, thought the purge removed them, on it.

k we good.

[buoyant lantern]

The Allen-Cahn equation is the result of L2 gradient flow on the Giznburg-Landau free energy for a given potential function

[prisma pelican]

wtf is the "standard approximation argument" in question here?

like, if Du_k converged to Du strongly

why does det Du_k converge to det Du?

I checked chatgpt and it gave this answer

and I'm sorry but like

???

amd I supposed to know this off the top of my head or smth?

[short barn]

Does strongly mean uniformly?

If so it’s just because det is continuous

[median forum]

It doesn’t mean uniform, but det being continuous is the reason

[prisma pelican]

no, as in it converges in L^q/n

[median forum]

It is useful to know that the determinant and inverse of a matrix are polynomial in the entries for stuff like this

That’s all this statement really is

[short barn]

Ok I guess you can also just say the determinant is a polynomial so it’s differentiable and hence Lipschitz

Or locally lipschitz

Like you don’t need to write the estimate like that, though I guess that’s more or less obvious from the proof of differentiable implies locally Lipschitz

[prisma pelican]

but Du_k doesn't necessarily converge to Du a.e.

[exotic lava]

🥀

[prisma pelican]

just in L^p

[median forum]

No but it’s good to make this clear to show that the constant depends on the norm of the thing

[exotic lava]

I mean kinda

that's just how polynomials work

[median forum]

You aren’t using a.e. convergence - you are presenting both sides of the identity as two continuous functions on your function space and then since they are equal on a dense set, they are equal on the whole space

That’s it in spirit, the details might be slightly different

[prisma pelican]

well, sure, but how do I show they're continuous

this should boil down to bounding both operators

but I don't know how to do that

this doesn't follow from det simply being continuous

[quick pagoda]

Bruh

[median forum]

Where in evans is this/what are the assumptions on p?

[quick pagoda]

Well it's not linear

[prisma pelican]

chreww

I didn't even realize

[prisma pelican]

page 449

[median forum]

This is why something like this is very helpful lol

[prisma pelican]

section 8.2

[quick pagoda]

Anyway, if it works when you converge pointwise a.e. then you always do

Because like, take sub sub sequences

[prisma pelican]

the formal lemma

[median forum]

Do you remember the Sobolev chain rule?

[prisma pelican]

the sobolev chain rule?

isn't that straight up just the chain rule?

[median forum]

Chapter 5 ex. 17

[prisma pelican]

well, now you need a bound to use DCT

[quick pagoda]

Pointwise estimate

[prisma pelican]

yes, well, sure, I'm just saying this may or may not be off the bounds of "standard approximation argument"

[quick pagoda]

Which sure seems doable given the integrability exponents involved

[quick pagoda]

Idk it seems pretty doable to me, the only trouble is write out a bound for the determinant thing

And that's kinda a polynomial

Lipschitz-y bounds and all that

[prisma pelican]

yea, I'm not sure how to get this using just lipschitz bounds ngl

tbh, I did get my answer

I know how to prove the equation holds for every funciton in W^1,q

I just don't see how it's as easy as yall make it seem

[glossy heart]

wth is the sobolev chain rule?

Oh just the chain rule

[median forum]

I'm referring to a chain rule/moser estimate

this type of bound

[frank tide]

what book/paper is this?

oh evan’s?

[velvet pier]

yes

[prisma pelican]

I don't understand this very well

how does this show the critical point is not a saddle point?

and I dont see how u(.,\lambda) for any lambda is different from u(., 0) = u(.)

this is such a weird theorem for me

ok, and the proof of this fact is fucking bonkers lmao

[median forum]

The other functions in the family are minimizers as well

In this sense

I think the point is that saddle points are extremal in some way

[prisma pelican]

wdym?

I'm not sure this is necessarily a saddle point

u(., lambda) could just be a family of local minimizers

that's in way the whole reason why I'm confused coz how does this show u(., lambda) is not a saddle point

coz this theorem (the only theorem here!) does not seem to answer the question posed at the intro of this subsection

[median forum]

The u_{\lambda} > 0 condition sort of means that the family gives some sort of nondegeneracy which allows you to write small variations as admissible ones

[median forum]

I mean that it is not a saddle. Saddle points are extremal in that they cannot be surrounded by such a family

[spiral geode]

from what book this is ?

i will just say that the cases of saddle points usually are excluded because of geometrique properties of I[.] ( convexity / coercivity )

[buoyant lantern]

Question: When interpreting some PDE (for example, heat eq) as a gradient flow over Wasserstein space, the W space is of measures, what are these measures, what do they define in the context of the PDE?

[ornate flare]

They represent the distribution of some conserved quantity, eg the distribution of heat/probability density of particles acting under pure brownian motion for ur heat equation

In general we can look at the actual energy functional that the gradient flow is minimizing and interpret the PDE as the continuity equation that results from how our prob mass moves across the domain to find its lowest energy configuration as efficiently as possible

So again for the heat equation our energy is just the negative entropy and so this forces our measure to spread out evenly

[prisma pelican]

evans

[prisma pelican]

sure, but that doesn't answer why the theorem is significant

I thought the whole point of this section is to show when a critical poiint is at least a local minimum but that's not what's been shown

[median forum]

Did you see this

[prisma pelican]

I did. I'm sorry, perhaps I'm not communicating myself properly

like, first of all, functions of the form w can be very small

indeed, I see no reason why u_lambda can't just be constant

and indeed that does seem to be possible

even if u is a saddle point

[median forum]

u_\lambda is the derivative w.r.t to lambda right?

[prisma pelican]

oh, I thought it was u(., \lambda)

hm

[median forum]

No I think it's the derivative, which gives a nondegeneracy condition on the family

[prisma pelican]

yea, you're right

so u_\lambda is positive everywhere

so at any point x, we can infer that u(x, \lambda) is within an epsilon neighborhood of u(x)?

[median forum]

I would phrase it as for any x, there is some $\epsilon > 0,$ for any $y$ s.t. $|u(x) - y| < \epsilon,$ there is some $\lambda$ such that $u(x,\lambda) = y$

[untold delta]

zeke

[prisma pelican]

ok, this makes sense

yea, ok, sure

but now I'm thinking isn't the condition u_lambda>0 super hard to meet?

maybe it's not in practice ig

[median forum]

I mean I think it is

[prisma pelican]

but like, if you take your typical saddle point in R^3, say x-y then the only critical point around (0, 0) is (0, 0)

[median forum]

I think the point is that it is very hard to show that critical points are minimizers

[prisma pelican]

perhaps, tho I would say evans made a very poor job of showcasing that point

I thought this was just like "here's an easy criterion to verify a crit point is a local minimizer"

but now my question is, why didn't evans just calculate the 2nd variation

[median forum]

I think it's supposed to be an extension of classical statement - which is the jacobi condition

[prisma pelican]

like it was hinted to in the beginning of the chapter

[median forum]

I mean you just have a weak solution - why can you differentiate further?

[prisma pelican]

you can make assumptions about regularity before you prove regularity ig

coz if you don't show your critical point is a local minimizer there really isn't a point of showing it's regular?

for the record, evans in thie subsection does assume u is at least C^1

and not just a weak solution

otherwise he wouldn't be able to swap u_\lambda x_i and u_xi lambda

[prisma pelican]

is it just me or is this like

way too weak

like, the euler-lagrange would just be uhh

like a quadratic on Du

well, not necessarily a quadratic but I suppose for any i, u_xi^a where a<= 2

for any a>2 this would fail 😭

[prisma pelican]

@river path am I getting this correctly

is it just hopeless to get some regularity theory on general lagrangians?

[sonic olive]

The purpose of this section is to provide a basic example, it’s not meant to cover general elliptic pde theory

But these are the most common types of functionals you will see anyway like -div(A nabla u) = f

But you can try to run the Degiorgi Nash Moser iteration on any Euler Lagrange equation you get

[prisma pelican]

inchresting

I literally just saw this for the first time 5 minutes ago on some random notes on some random PDE course website lmao

they don't seem to be covered in evans tho

are these a standard technique?

[sonic olive]

Yes but you would need to find a book on elliptic/parabolic pde

It is probably the most famous/important result in divergence form elliptic/parabolic pde

Essentially they say that the mean value inequalities for harmonic functions work for divergence form equations

The iteration schemes also work when q is not 2 so you can run them on p-Laplace type equations as well

[prisma pelican]

inchresting

[prisma pelican]

currently, I'm not super interested in learning more about 2nd order PDEs tbh

out of curiosity

what's the usual "next step" after evans

[sonic olive]

You decide which type of pde you like and find a more advanced treatment of that subject

[prisma pelican]

makes sense lmao

and I'm assuming after that you just start reading papers?

[sonic olive]

Yeah you can but evans doesn’t really prepare you yet to read papers

But yeah after you maybe do one more book you should be good

[prisma pelican]

an exciting propspect

or perhaps a dreadful one

can't really tell from my vintage point

currently, after evans I think I wanna go into optimal transport and gradient flows

and hopefully after 1-2 books there I'll be able to read papers too

[sonic olive]

Nice, I have no background in this so I wouldn’t know lol

[prisma pelican]

but yea, gradient flows do intersect PDEs heavily so I might shift back

[sonic olive]

Is monge ampere related to optimal transport somehow?

Some masters student told me this a couple years back but I don’t really see the connection

[prisma pelican]

monge ampere came from optimal tranport originally

they're basically the PDE satisfied by the optimal transport map

but they're more like PDEs for the sake of optimal transport

Of course, there's the other way around

wherein optimal tranposrt allows you to model evolution PDEs as some sort of geodesic or a gradient flow in the space of probability measures

first instance of this was due to otto felix, he did it for porous medium equations

it's pretty cool

[buoyant lantern]

thanks! so the space where we are decaying the energy functional over is the "possible distributions" of some continuous density?

im trying to train neural networks to perform gradient descent, though right now i havent tackled the wasserstein space

[ornate flare]

Yes

[ornate flare]

Is there a specific NN structure ur trying to optimize if I may ask?

[buoyant lantern]

actually its just an MLP, a physics-informed MLP specifically that, at the n-th training step has to predict a function (neural field) that satisfies the PDE (contunuity eq.): the twist is that I force the NN to monotonically decay the energy functional, "ensuring" (really moreso, nudging) convergence

[ornate flare]

Sounds interesting. Do u just kinda slap the energy functional onto the loss function with a Lagrange multiplier?

[buoyant lantern]

Thats what normal PINNs do (except they slap the PDE residual into the loss), I may do that but I want there to be a "global clock", and at the n-th "time"-step, the NN is trained to satisfy the PDE, and the constraint is that the energy of the predicted solution has to be lower than that of the n-1th "time"-step

[tropic oxide]

What is this perspective usually used for? Does it help you get new estimates or help with regularity/existence

[prisma pelican]

that's a good question

I have no idea

I'm yet to read otto's paper

but it's supposedly quite successful

it spawned the whole JKO scheme thing

which is quite a hot research topic rn

[tropic oxide]

Based on just reading this introduction it seems like yes, it can give new estimates https://cvgmt.sns.it/media/doc/paper/7085/lecturenotes Chania Santambrogio.pdf that’s pretty cool!

[buoyant lantern]

as I recall from a talk, looking at a PDE as gradient flow on wasserstein space is great because a new definition of distance => new topology => things that were not convex in R^n may be convex under our new definition of distance. Which i guess would give some good guarantees on existence/convergence as you said above. not to mention we can use new discrete schemes like JKO

ah and i just saw the PDF is about JKO!

im trying to learn it too

[analog zephyr]

How do we compute what a^{ij}(x) is if we don’t know what u is?
Like, sure, it’s a solution to this equation
But how do we know the smallest eigenvalue goes to 0?
Like it’s definitely positive, but not necessarily approaching 0

[median forum]

Typically in the case when you have a^{ij} = a^{ij}(x,u,Du) you want ellipticity bounds independent of x, u, Du - though maybe there are cases where you can assume some weak form and then bootstrap a stronger condition

So here it is zero as that is just the only bound you can prove a priori

[quick pagoda]

I hate these notations, but like uhhh write out the Nabla • thing a bit more, and use that the divergence is zero to see that that eigenvalue drops

Since that minimal surface thing constrains those a_ij

I don't know a good slick intuitive argument offhand to why this works out tho

[quick pagoda]

This is very nice though

[digital ibex]

Im trying to derive the variational form of Possion’s equation, but I’m missing a 1/2 factor and I can’t figure out why

\begin{align}
-\int u(x)v(x) dx &=\int f(x)v(x)dx\
\text{let } v(x) &= u(x)\
\implies \int \left(u’(x)\right)^2 dx&= -\int f(x)u(x)dx
\end{align}

[untold delta]

KySquared

[digital ibex]

There should be a 1/2 on the LHS on line 3 but idk how that comes up

[quick pagoda]

Are you sure (1) is right

[median forum]

What's up cool people

Where do you all recommend learning di giorgi-nash from

bonus points for parabolic stuff as well

[quick pagoda]

Imbert has some notes that go over it in some linear settings but for elliptic, parabolic, and kinetic

[median forum]

Looks excellent thanks!

[quick pagoda]

Specifically linear stuff so no iteration in nonlinear stuff, but I dunno anywhere else that covers kinetic

In that same flavor anyway

[tropic oxide]

I also like “An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs” by Giaquinta and martinazzi, they have 3 or 4 proofs (de Giorgi, moser iteration, and some others)

Only for elliptic equations though

[quick pagoda]

Includes system-y things though

[worn badger]

Second

[river path]

[wind mortar]

existence problems?

does 0 exist

[river path]

Yes. By elliptic existence/uniqueness theorems, there exists a solution to $\begin{cases} \Delta u = u & \textrm{on } B_1 \ u = -u & \textrm{on } \partial B_1 \end{cases}$. We define $0$ to be any value in the image of $u$. We claim that $0$ is well defined.

[limpid slate]

yo this is new too

[wind mortar]

bruh

isn't that cyclical

[river path]

fine

fixed

[untold delta]

ryc

[river path]

helmholtz moment

idk if this is actually true

i dont recall

[tired hollow]

I'm a bit confused here, how can the solution to the BS PDE be AS?

Am I missing some obvious algebraic trick to force it to be true? I don't get how you can turn Kexp(-r(T-t)) into that

?

@atomic maple aren't you the resident math finance guy, why haven't you forced them to make a channel for you yet

[river path]

what does it mean for the underlying to not pay out dividends?

[tired hollow]

The stock S has no dividend payments

[atomic maple]

i was told that's what advanced probstats is for

[tired hollow]

So won't impact the option price

[tired hollow]

lame 😭

[atomic maple]

:/

[tired hollow]

Is this Q really obvious? It's the first exercise in my notes but I don't get it

How can the option price be A * S?

[atomic maple]

isn't the black scholes equation $$\pdv {V}{t} + \frac 12 \sigma^2 \pdv[2]{V}{S} + rS \pdv{V}{S} - rV = 0$$

[untold delta]

Steakanator

[tired hollow]

Missed S^2 by the 1/2 but yea

[atomic maple]

well in this case it doesn't matter so

[tired hollow]

huh?

[atomic maple]

that term is 0

[tired hollow]

Sure since AS partial differentiated against S twice

The main issue is partial AS against t

I assume you are proposing just set V = AS and see what happens?

[atomic maple]

quite so

[tired hollow]

$$A\pdv {S}{t} + rAS - rAS = 0$$

[untold delta]

Sam W

[tired hollow]

S=S(t) so not sure how to tackle that partial

$$\pdv {S}{t}= 0$$

[untold delta]

Sam W

[tired hollow]

[atomic maple]

yeah that's about it

[tired hollow]

How does that show that it is a solution though?

or well hmm

guess I was just overthinking it

[atomic maple]

it's a solution if it satisfies the pde

[soft summit]

why does the first equation show the sum is 0

[dawn lava]

The summands are the coordinates of the gradient

And the gradient is zero, so all the summands are zero?

(I'm sleepy I might've chain rule'd wrong in my head)

What's the context btw?

[soft summit]

But they're multiplied by partials of phi

this is about envelopes for nonlinear 1st order

evans page 94

[dawn lava]

Isn't that the chain rule?

You're composing with u with (Id,phi)

The first term of the rhs of the first equality is what comes out of Id, the second term what comes out of phi?

[soft summit]

yes but why is the second term 0

[dawn lava]

I think that term might be (10) by definition pretty much?

Sorry I'm really sleepy and I don't have paper at hand 😓

Let me see if I can figure it out

[dawn lava]

(12)*

Ok, D_a is just taking the gradient on the A part of the coordinates

So if you just restrict to that part of the space, the summand is pretty much just (12) expanded by using the chain rule I think

It's a bit confusing cause one would expect the non-A coordinates to not affect it, but it does cause now a is phi(x), ie depends on the first few coordinates

[soft summit]

but phi(x) is a

[dawn lava]

The a in D_a is not phi(x) fwiw

It just vaguely represents the fact that you're considering the gradient restricted to A

[soft summit]

so it's still gradient wrt x

[dawn lava]

It's a part of it

Not the entire gradient

Like in the heat and wave equations but more general, if that makes sense

When you ignore the t part of the gradient, you're similarly ignoring the non-A part here

Thinking about this proof with the heat equation might actually be a good way to get some intuition for it I guess

Like, in heat equation terms, the hypothesis of this theorem would be somthing like saying the temperature is constant across all space for fixed t, but depends on t

I think

[soft summit]

ok I think I'm getting the idea

tyvm

[dawn lava]

Basically, the second summand is evaluating the A-gradient at the vector D_x phi(x_i)

Well not exactly evaluating it

More like taking the inner product?

But since the gradient is zero that will be zero too

[verbal nebula]

is this out of evans?

[soft summit]

yes

[verbal nebula]

I skipped all the characteristic and envelope stuff

[soft summit]

I need to do a presentation for pde class

and the prof skipped this too

[verbal nebula]

I can tell you what not to do

Anything by Thomas Liggett

I chose a rather technical paper to present on, and it was just

Not good

[soft summit]

👀
well she said we can just pick whatever's in the book that she hadn't talked about in class
or something else
and specifically anything that isn't algebra lol
so I'm thinking something about Hamilton-Jacobi

[verbal nebula]

Sobolev spaces are really cool

So is regularity theory

There's also lots of fun stuff in calculus of variations

Like energy methods

Taking minimizers

[soft summit]

we went through sobolev tho

[verbal nebula]

fk

What chapters of the book did you do?

[soft summit]

yeah variational stuff is another one I'm thinking about

125 currently at 6

[verbal nebula]

Ah ok, so that's all about weak formulations, Lax-Milgram

Regularity, etc.

That stuff is kinda fun

[astral vine]

Gorgeous channel

[verbal nebula]

I know right

No more getting bombarded with freshman calculus epsilon deltas

[astral vine]

That's exactly the point

[verbal nebula]

I need to get back to work in Evans. I only solved like 2 or 3 problems in chapter 6

and it's been a good while

It's a little annoying to have harmonic and PDEs in different channels since I'm at the intersection of both

But eh

[dawn lava]

I think they're cursed tbh

As I was reading that section of Evans I was like "ok sure I've got an intuition here" but all the while I'd just been thinking about functions from R to R

And ohhh boy will that lead you astray

[verbal nebula]

Fun fact, my first PDE class started with sobolev spaces

[dawn lava]

Evans has just two cute examples to leave you in utter horror at what a function in Sobolev space can look like, and it's more than enough

[astral vine]

A nice corollary, it's that one can send its Functional Analysis questions, or Applied Functional Analysis questions in both channel depending on the goel

[verbal nebula]

how do you type lmao in french

rmc