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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.

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Solutions to the wave equation on non orientable surfaces like a mobius strip

Given a mobius strip, what do the solutions of the wave equation look like qualitatively? How do they differ from solutions on the equivalent strip glued together as a cylinder? Any refs, ...
6
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1answer
500 views

Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)

The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...
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5answers
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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 ...
27
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5answers
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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, ...
4
votes
3answers
914 views

Harmonic Functions

Suppose f: RxR -> R is has continuous partial derivatives and 4* f(x,y)=f(x+del,y+del)+f(x-del,y+del)+f(x-del,y-del) + f(x+del,y-del) for all (x,y) in RxR and all del in R. I dont believe that f is ...
3
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0answers
175 views

Generalizations of group algebras for arbitrary manifolds?

In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take ...
6
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3answers
962 views

Green's formula for nonorientable manifolds

Usually in differential geometry one proves the Stokes theorem and then obtains divergence theorem and Green's formulas as corollaries. However, divergence theorem is also valid for nonorientable ...
16
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6answers
6k views

Can an integral equation always be rewritten as a differential equation?

Given an integral equation is there always a differential equation which has the same (say smooth) solutions? It seems like not but can one prove this in some example? Edit: Naively I'm hoping for ...
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3answers
2k views

Surveys on Navier Stokes Equations and its physical implications

Hi, I'm a beginning graduate student, and I'm interested in learning more about Fluid Mechanics and, in particular, the Navier stokes Equations. I would like to know: are there are some sort of free ...
2
votes
1answer
432 views

Identifying the nonlinear parabolic PDE $u_t = (u^2)_{xx}$.

A friend of mine in the department needs to know if the following PDE has been extensively studied $$ u_t = (u^2)_{xx}$$ Or more generally, replacing the square by any function of $u$. One would like ...
4
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4answers
3k views

The characteristic (indicator) function of a set is not in the Sobolev space H1

Is it true that the characteristic (indicator) function of a subset of Euclidean space with finite positive measure is never in the Sobolev space $H^1 = W^{1,2}$ And if so, what is the best/...
3
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1answer
3k views

Is there a general method for determing the domain of dependence of (higher-order) PDEs?

Is there a general method for determining the domain of dependence of (higher-order) PDEs? I would be plenty happy with a reference to a paper or textbook I could look at; despite much reading around,...
4
votes
3answers
1k views

Numerical instability using only Heun's method on a simple PDE.

I'm trying to simulate the evolution of the Wigner function (a pseudo probability distribution over phase space) for a point particle moving in a chaotic potential. I'll provide background first, as ...
5
votes
0answers
136 views

Regularity of reflection coefficients (or more generally the scattering transform)

Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$. We define the ...
6
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2answers
517 views

What are the interesting cases of the generalized Korteweg-de Vries equation?

The generalized Korteweg-de Vries equation is $u_t + u_{xxx} + (u^p)_x=0$ for integer $p$. (The original Korteweg-de Vries equation is the case $p=2$.) I need to understand solutions for $p=1$, ...
4
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3answers
1k views

Connected components of space of maps between two manifolds

Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$? Specifically, I'm thinking of the ...
5
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1answer
422 views

Sign-Gordon Equation

What can be said and done about the "SIGN-Gordon equation"? $$\varphi_{tt}- \varphi_{xx} + \text{sgn}(\varphi) = 0.$$ It came up here.
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vote
1answer
266 views

References for weak ellipticity

There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as ...
1
vote
1answer
544 views

Entropy of Markov processes

Consider a Markov process $X_t$ with generator $L$ and invariant distribution $\pi$, whose distribution at time $t$ is given by $\pi(t,dx)=\phi(t,x) \pi(dx)$ - in other word, $\phi(t,x)$ is the ...
16
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5answers
2k views

“Physical” construction of nonconstant meromorphic functions on compact Riemann surfaces?

Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic ...
18
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2answers
2k views

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 ...
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2answers
930 views

Rellich-Necas identity

I am looking for a book/paper which has the proof of the Rellich-Nicas identity. [EDIT by Yemon Choi] It seems that what was meant is "the Rellich-Necas identity", although the original poster hasn't ...
7
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3answers
804 views

Newlander-Nirenberg for surfaces

Quite a long ago, I tried to work out explicitly the content of the Newlander-Nirenberg theorem. My aim was trying to understand wether a direct proof could work in the simplest possible case, namely ...
11
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1answer
819 views

Prescribing Gaussian curvature

Let $K(r)$ be the piecewise function                            &...
2
votes
4answers
4k views

Undergraduate Derivation of Fundamental Solution to Heat Equation

It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
17
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3answers
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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 ...
7
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3answers
834 views

Solutions to a Monge-Ampère equation on the simplex

Let $\Delta_k$ be the k-simplex and $\mu$ a non-negative measure over $\Delta_k$. I want to know if there exists a function $u : \Delta_k \to \mathbb{R}$ such that $u$ is convex, $u(e_i) = 0$ for all ...
11
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2answers
2k views

Hypoellipticity of square root of laplacian

It is a well known result (sometimes called the Weyl lemma) that the laplacian in $\mathbb{R}^n$ is hypoelliptic, i.e. if $f$ is a distribution s.t. $\triangle(f)$ is smooth in an open set, than $f$ ...
6
votes
1answer
573 views

primitive of an exact differential form with special properties

We were working on a smoothing problem and ran across the apparently simple following question: X is a triangulated smooth manifold of dimension $n$, and $\alpha$ is an exact differential form of ...
23
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2answers
2k views

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 ...
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votes
9answers
16k views

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 ...
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2answers
704 views

Integrable solutions to an elliptic PDE on divergence form have a definite sign

Let $f\colon\mathbb{R}^n\to\mathbb{R}^n$ be a smooth, bounded vector field. Further, let $u\colon\mathbb{R}^n\to\mathbb{R}$ satisfy $$-\Delta u=\operatorname{div}(fu).$$ If $u\in L^1(\mathbb{R}^n)$, ...
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8answers
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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 ...
16
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6answers
5k views

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 ...
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4answers
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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 ...
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3answers
824 views

Equations for Integrable Systems

So, let's say we have a symplectic variety over $\mathbb{C}$, $M$, of dimension $2n$, and $f_1,\ldots,f_n$ Poisson commuting functions with $df_1\wedge\ldots\wedge df_n$ generically nonzero. Further ...
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votes
1answer
145 views

Transformation from domains to half-spaces

In a paper I read, an elliptic boundary value problem on a bounded domain D x (0,T) is solved by first transforming it in a set of equations on half-spaces R^n and then applying partial Fourier ...
9
votes
2answers
3k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
2
votes
3answers
830 views

Sobolev norms of eigenfunctions

Let D be a domain in R^n, and let f be an eigenfunction of the Laplacian with Dirichlet boundary condition with eigenvalue $\lambda$. Assume that f has L^2 norm 1. I want to know if I can say anything ...
3
votes
3answers
1k views

Error analysis of implicit functions

I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms): $$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
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votes
7answers
11k views

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....
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2answers
666 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ansätze do you know ...
14
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6answers
1k views

Can the “physical argument” for the existence of a solution to Dirichlet's problem be made into an actual proof?

Caveat: I don't really know anything about PDEs, so this question might not make sense. In complex analysis class we've been learning about the solution to Dirichlet's problem for the Laplace ...
4
votes
1answer
4k views

Transformation of the Black-Scholes PDE into the diffusion equation - shift of coordinate system

The aim of transforming the Black-Scholes PDE is of course to find a form where an relatively easy solution exists. Most of the steps seem to be straightforward - please use this reference: https://...
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2answers
2k views

Solutions to the diffusion equation

When it comes to solving the heat diffusion equation u_t=u_xx the two most important solutions are a) a combination (sum) of sin-terms to resemble the function of the initial condition (that is ...
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2answers
2k views

Eigenvalues of Laplacian

What's the most natural way to establish the asymptotics of $\Delta$ on a compact Riemannian manifold $M$ of dimension $N$? The asymptotics should be $$ \#\{v < A^2\} = \mathrm{const}\ast\mathrm{...