Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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121
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5answers
9k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
92
votes
8answers
47k views

Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations? [closed]

Moderator Notice: I am unilaterally closing this question for three reasons. The discussion here has turned too chatty and not suitable for the MSE framework. Given the recent pre-print ...
51
votes
1answer
5k views

What is the solution to Nash's problem presented in “A Beautiful Mind”?

I was watching the said movie the other night, and I started thinking about the equation posed by Nash in the movie. More specifically, the one he said would take some students a lifetime to solve ...
48
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8answers
20k views

Good 1st PDE book for self study

What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic ...
47
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5answers
2k views

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and ...
44
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5answers
11k views

Is the problem that Prof Otelbaev proved exactly the one stated by Clay Mathematics Institute?

Recently, mathematician Mukhtarbay Otelbaev published a paper Existence of a strong solution of the Navier-Stokes equations, in which he claim that he solved one of the Millennium Problems: existence ...
42
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1answer
6k views

Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic

what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order. So far, I have ...
39
votes
2answers
3k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
32
votes
8answers
2k views

Partial differential equations in “pure mathematics”

One thing I have noticed about PDEs is that they come from Mathematical Physics in general. Almost all the equations I see in Wikipedia follow this pattern. I can't help wondering whether there are ...
28
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6answers
2k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
27
votes
2answers
1k views

PDEs on manifold: what changes from Euclidean case?

I know some PDE theory for nice open domains in $\mathbb{R}^n.$ I want to know what the changes are when I switch to other domains like manifolds. For example, do things like Poincare's inequality ...
26
votes
2answers
913 views

Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why. ...
26
votes
3answers
1k views

Non-linear partial differential equation

I would like to find out if there is any specific method -apart from numerics- for finding solutions of a non-linear PDE of the form $$\nabla \times \mathbf{A} = \pm\lambda\mathbf{A} \tag{1}$$ under ...
25
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2answers
2k views

Intuitive explanation of the difference between waves in odd and even dimensions

Motivation: In odd dimensions, solutions to the wave equation: $u_{tt}(x,t)=\nabla u(x,t)$, $u_t(x,0)=0$, $u(x,0)=f(x)$, ($t\geq 0, x\in \mathbb{R}^n$) have the nice property that the value of ...
24
votes
2answers
851 views

Why do odd dimensions and even dimensions behave differently?

It is well known that odd and even dimensions work differently. Waves propagation in odd dimensions is unlike propagation in even dimensions. A parity operator is a rotation in even dimensions, ...
23
votes
2answers
1k views

Applications of Pseudodifferential Operators

I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
22
votes
1answer
745 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
21
votes
1answer
674 views

Was Euler right?

We have a differential equation $$ y + y' = f(x) $$ and assume $f$ is infinitely differentiable. And we want to find particular solution. Then,I set $$ y_p = f(x)-f'(x)+f''(x)...., $$ i.e., ...
20
votes
2answers
298 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
17
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2answers
1k views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
17
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2answers
1k views

PDEs on Manifolds

I am wondering if there is a general coordinate-independent way to define a Partial Differential Equation on a Smooth manifold. It is definitely true that in each coordinate neighborhood you could ...
17
votes
4answers
1k views

The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

I am interested in solving the following biharmonic eigenvalue problem. $$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\ ...
17
votes
2answers
2k views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
17
votes
3answers
2k views

What is the purpose of computing the eigenvalue of a PDE problem?

I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one). But when I solve ...
17
votes
4answers
801 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
17
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1answer
453 views

Dual Bases: Finite VS Infinite Dimensional Spaces

Motivation I know that in finite dimensional spaces like a $n$ dimensional Euclidean space $\mathbb{E}^n$, for every basis $G=\{g_1,g_2,...,g_n\}$ we can define a dual basis $G'=\{g^1,g^2,...,g^n\}$ ...
16
votes
1answer
816 views

Stochastic interpretation of Einstein equations

Einstein's theory of gravitation, general relativity, is a purely geometric theory. In a recent question I wanted to know what the relation of Brownian motion to the Helmholtz equation is and got a ...
16
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0answers
356 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
15
votes
3answers
1k views

Do discontinuous harmonic functions exist?

A function, $u$, on $\mathbb R^n$ is normally said to be harmonic if $\Delta u=0$, where $\Delta$ is the Laplacian operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. So obviously, ...
15
votes
4answers
566 views

Roadmap to SPDEs

I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory. I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and ...
15
votes
4answers
513 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
15
votes
5answers
4k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
15
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2answers
6k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
14
votes
3answers
574 views

Connections between K-Theory and PDEs?

I've recently spent some time learning (the very basics of) K-theory for $C^*$-algebras and topological K-theory. Actually, my main fields of interest are PDEs and related topics, in particular ...
14
votes
4answers
25k views

What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best and why? Advantage and disadvantage of them?
14
votes
3answers
5k views

Proof of the Gauss-Green Theorem

I can't seem to find any references that gives a proof of the Gauss-Green theorem: Let $U\subset\mathbb{R}^{n}$ be an open, bounded set with $\partial U$ being $C^1$. Suppose $u\in ...
14
votes
2answers
519 views

Reversing the Ricci flow

Suppose $S$ is a closed, oriented surface (2-manifold) embedded in $\mathbb{R}^3$, which inherits the metric from $\mathbb{R}^3$, so that distances are measured by shortest paths on the surface. If ...
14
votes
0answers
282 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
13
votes
1answer
1k views

Are a weak derivatives and distributional derivatives are different?

For simplicity, given a real function $f\in L^1_{loc}(\Omega)$, we define both weak or distributional derivatives by $\int f'\phi = - \int f \phi'$ for all test functions $\phi$. Now, take $\Omega = ...
13
votes
2answers
1k views

viscosity solution vs. weak solution

I am confused between the two. Is one a subset of the other or they are the same/completely different notions? Say I have an euqaiton $u_t=\mathcal{L}u$ for an elliptic operator $\mathcal{L}$ with bad ...
13
votes
1answer
162 views

Interpretation of an integral transform from the wave equation to the heat equation

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ ...
13
votes
2answers
490 views

Elliptic regularity in Sobolev spaces of negative order

I am having some trouble with Sobolev spaces of negative order. More precisely I am considering the space $W^{-1,p}(\mathbb{R}^2),$ considered as 'the' dual space of $W^{1,q}(\mathbb{R}^2).$ Question ...
13
votes
0answers
139 views

Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE ...
12
votes
2answers
5k views

Energy norm. Why is it called that way?

Let $\Omega$ be an open subset of $\mathbb{R}^n$. The following $$\lVert u \rVert_{1, 2}^2=\int_{\Omega} \lvert u(x)\rvert^2\, dx + \int_{\Omega} \lvert \nabla u(x)\rvert^2\, dx$$ defines a norm on ...
12
votes
2answers
2k views

How to define weak solution for an elliptic PDE with non-zero Dirichlet boundary condition?

For homogeneous Dirichlet boundary condition, for example $$ \left\{\!\! \begin{aligned} &-\Delta u+c(x)u=f(x),x\in\Omega\\ &u|_{\partial\Omega}=0 \end{aligned} \right. $$ The weak ...
12
votes
6answers
1k views

Good PDE books for a graduate student?

I am now a graduate student in mathematics, and I really want to learn more about PDE. I would say I have a very solid foundation in soft analysis, including functional analysis and harmonic analysis, ...
12
votes
2answers
2k views

A Problem in Evans' PDE

Problem 7 in §6.6 states as follows: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$ where ...
12
votes
2answers
236 views

A technical relation

I have encountered the following interesting technical relation. $$ \pi^2 = \inf_{x \in \mathcal{D}(0,1) \setminus\{0\}} \frac{\int_0^1 |x'(s)|^2 \, \text{d}s}{\int_0^1 |x(s)|^2 \, \text{d}s}$$ ...
12
votes
1answer
541 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
12
votes
2answers
1k views

classical solutions of PDE with mixed boundary conditions

Nowadays people usually consider PDEs in weak formulations only, so I have a hard time finding statements about the existence of classical solutions of the Poisson equation with mixed ...