# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2,558
questions

**104**

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**16**answers

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### Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...

**94**

votes

**8**answers

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### What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...

**65**

votes

**13**answers

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### Counterexamples in PDE

Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their ...

**64**

votes

**9**answers

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### Why can't there be a general theory of nonlinear PDE?

Lawrence Evans wrote in discussing the work of Lions fils that
there is in truth no central core
theory of nonlinear partial
differential equations, nor can there
be. The sources of partial
...

**64**

votes

**2**answers

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### Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...

**60**

votes

**7**answers

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### What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....

**51**

votes

**2**answers

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### Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...

**49**

votes

**9**answers

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### Motivation for and history of pseudo-differential operators

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds). Which prominent example problems lead you to work with pseudo-differential ...

**46**

votes

**6**answers

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### Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation
I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...

**40**

votes

**1**answer

4k views

### Unconditional nonexistence for the heat equation with rapidly growing data?

Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: [...

**37**

votes

**2**answers

2k views

### Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...

**32**

votes

**5**answers

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### Why is symplectic geometry so important in modern PDE ?

First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds ...

**32**

votes

**4**answers

3k views

### Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...

**32**

votes

**1**answer

3k views

### Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...

**31**

votes

**12**answers

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### Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)
I am an undergrad in math ...

**31**

votes

**1**answer

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### A long-lasting conjecture of Pólya & Szegő

There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) which is as follows:
"Of all $n$-gons of a fixed area, the regular ...

**30**

votes

**5**answers

2k views

### Explicit Eigenvalues of the Laplacian

Let $(M,g)$ be a compact manifold without boundary.
Question: For which $(M,g)$ are the eigenvalues of the Laplace operator on functions explicitly known?
An important example is the $n$-sphere ...

**30**

votes

**5**answers

2k views

### How to define a differential form on a fractal?

It is well known how to construct a Laplacian on a fractal using the Dirichlet forms (see e.g.
the survey article by Strichartz). This implies, in particular, that a fractal can be "heated", i.e. one ...

**28**

votes

**5**answers

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### H-principle and PDE's

According to Wikipedia: "In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally partial differential relations ...

**28**

votes

**5**answers

3k views

### Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes?

Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is turbulent in the spacetime region $U$"?
And if such a definition exists, ...

**28**

votes

**1**answer

2k views

### The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_{...

**27**

votes

**4**answers

3k views

### Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...

**27**

votes

**3**answers

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### “Wild” solutions of the heat equation: how to graph them?

It has long been known that the Cauchy initial-value problem for the
classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't
have unique solutions, without additional assumptions. In ...

**27**

votes

**1**answer

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### Jet bundles and partial differential operators

A geometric way of looking at differential equations
In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...

**26**

votes

**2**answers

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### Simulating Turing machines with {O,P}DEs.

Qiaochu Yuan in his answer to this question recalls a blog post (specifically, comment 16 therein) by Terry Tao:
For instance, one cannot hope to find an algorithm to determine the existence of ...

**26**

votes

**1**answer

2k views

### What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities:
If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying $\frac{...

**25**

votes

**18**answers

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### PDEs as a tool in other domains in mathematics

According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural ...

**25**

votes

**1**answer

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### The origin of Discrete `Liouville's theorem'

It is known that discrete Liouville's theorem for harmonic functions on $\mathbb{Z}^2$ was proved by Heilbronn (On discrete harmonic functions. - Proc. Camb. Philos. Soc. , 1949, 45, 194-206).
If ...

**22**

votes

**8**answers

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### Applications of PDE in mathematical subjects other than geometry & topology

Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau ...

**22**

votes

**4**answers

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### Convergence of finite element method: counterexamples

There are many known results proving convergence of finite element method for elliptic problems under certain assumptions on underlying mesh [e.g., Braess,2007]. Which of these common assumptions are ...

**22**

votes

**5**answers

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### Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...

**22**

votes

**4**answers

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### What is the relationship between various things called holonomic?

The following things are all called holonomic or holonomy:
A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...

**22**

votes

**1**answer

664 views

### Eigenvalues of Laplace operator

Assume that $(M,g)$ is a Riemannian manifold.
Is there any relation between the sequence of eigenvalues of Laplace operator acting on the space of smooth functions and the sequence of eigenvalues of ...

**21**

votes

**7**answers

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### Applications of Microlocal Analysis?

What examples are there of striking applications of the ideas of Microlocal Analysis?
Ideally i'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/...

**21**

votes

**6**answers

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### Square roots of the Laplace operator

In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...

**21**

votes

**2**answers

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### Does Ricci flow with surgery come from sections of a smooth Riemannian manifold?

More precisely, is Ricci flow with surgery on a 3-dimensional Riemannian manifold M given by the "constant-time" sections of some canonical smooth 4-dimensional Riemannian manifold?
There would be a ...

**21**

votes

**3**answers

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### Does elliptic regularity guarantee analytic solutions?

Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard ...

**21**

votes

**0**answers

520 views

### What is a Green's function in the language of $\mathcal{D}$-modules?

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying ...

**19**

votes

**4**answers

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### When to use more exciting function spaces than ordinary Sobolev spaces?

In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.
For example, Sobolev spaces $L^2(0,T;H^...

**19**

votes

**4**answers

3k views

### Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...

**19**

votes

**3**answers

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### Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding.
Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...

**19**

votes

**4**answers

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### What is an “integrable hierarchy”? (to a mathematician)

This is one of those "what is an $X$?" questions so let me apologize in advance.
By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so ...

**19**

votes

**2**answers

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### Compact Embeddings of Sobolev Spaces: A Counterexample Showing The Rellich-Kondrachov Theorem Is Sharp

Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...

**18**

votes

**2**answers

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### Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes?

As I understand it, Lions and DiPerna demonstrated existence and uniqueness for the Boltzmann equation. Moreover, this paper claims that
Appropriately scaled families of
DiPerna–Lions ...

**18**

votes

**3**answers

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### History of fundamental solutions

I have a few questions on the history of PDE.
Who first wrote down the formula for the solution of the Cauchy problem for the heat equation involving the heat kernel? I have seen it called Poisson's ...

**18**

votes

**1**answer

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### Image of the trace operator

It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...

**17**

votes

**11**answers

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### Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...

**17**

votes

**6**answers

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### PDE on manifolds

I am currently in a PDE course where one of the requirements is to present a paper in PDE. I am wondering if anyone can suggest an early (read foundational, first introductory) paper talking about PDE ...

**17**

votes

**3**answers

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### A Hölder continuous function which does not belong to any Sobolev space

I'm seeking a function which is Hölder continuous but does not belong to any Sobolev space.
Question: More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \...

**17**

votes

**3**answers

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### Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its role in Getzler's work on Atiyah-Singer?

I'm in the process of understanding the heat equation proof of the Atiyah-Singer Index Theorem for Dirac Operators on a spin manifold using Getzler scaling. I'm attending a masters-level course on it ...