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

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103
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5answers
7k 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 ...
87
votes
8answers
40k 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 ...
44
votes
1answer
3k 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 ...
41
votes
5answers
10k 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 ...
38
votes
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 ...
30
votes
7answers
10k 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 ...
29
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 ...
27
votes
1answer
4k 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 ...
26
votes
2answers
2k 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?
26
votes
2answers
882 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 ...
25
votes
6answers
1k 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 ...
25
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 ...
22
votes
2answers
498 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. ...
21
votes
2answers
506 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, ...
20
votes
2answers
847 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 ...
19
votes
2answers
869 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, ...
19
votes
1answer
622 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., ...
19
votes
1answer
582 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 $u=u(x,y)$: ...
15
votes
2answers
151 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 exponenent allows for convenient ...
14
votes
2answers
912 views

Why are harmonic functions called harmonic functions?

Are they related to harmonic series in any way? Or something else? Wikipedia didn't help.
13
votes
3answers
482 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 ...
13
votes
4answers
400 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 ...
13
votes
3answers
1k 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 ...
13
votes
2answers
877 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 ...
12
votes
3answers
857 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, ...
12
votes
4answers
365 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 ...
12
votes
2answers
3k 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
915 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 ...
12
votes
1answer
419 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
1answer
578 views

The Helmholtz equation: How prove this $T\psi{(x)}\in\Omega$.

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...
11
votes
2answers
895 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 ...
11
votes
2answers
199 views

A technical relation

I have encountered the following interesting technical relation. $$ \pi^2 = \inf_{x \in \mathcal{D}(0,1) \setminus\{0\}} \frac{\int_0^1 |x'(s)|^2 \, \text{d}s}{\int_0^1 |x(s)|^2 \, \text{d}s}$$ ...
11
votes
1answer
124 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
11
votes
2answers
830 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 ...
11
votes
1answer
440 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
11
votes
2answers
404 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 ...
11
votes
1answer
494 views

Book recommendations for self-study at the level of 3rd-4th year undergraduate

I have only recently discovered an interested in mathematics and I could only take a year off work to be back at school. Needless to say, for financial reasons (couple of mortgages) I will need to ...
11
votes
2answers
219 views

Positivity of principal eigenvalue for $L\phi=-\triangle \phi + \nabla \cdot( u \phi )$

EDIT: This question is still unresolved as of April 18. The two answers provide useful work in the right direction, but neither resolves the question. A counterexample should have $u \in ...
11
votes
0answers
265 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 ...
11
votes
1answer
561 views

Hille Yosida theorem application

Disclaimer: pretty long and specific (contraction semi groups involved). I have fourth order parabolic equation $$ u_t + \Delta^2 u = 0 $$ on $U_T = U \times [0,T]$. $U \subset \mathbb{R}^m$ is a ...
10
votes
2answers
249 views

Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
10
votes
3answers
2k views

The mathematics of music - why sine waves?

Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal. But what ...
10
votes
5answers
2k 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.
10
votes
2answers
618 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
10
votes
5answers
345 views

What is the motivation behind a product solution?

Let's consider the simple differential equation: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$ And let's assume we have some regular homogeneous boundary conditions ...
10
votes
2answers
335 views

Mean value property with fixed radius

I will focus on the real line. Let $f$ be a smooth function on $\mathbb{R}$, if $\forall x\in\mathbb{R}, r>0$, $$\frac{f(x-r)+f(x+r)}{2}=f(x),$$ we say that f has the spherical mean value property ...
10
votes
1answer
421 views

Why no trace operator in $L^2$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either ...
10
votes
2answers
251 views

Non-ellipticity of Yang-Mills equations

Let $D=\text{d}+A$ be a metric connection on a vector bundle with curvature $F=F_D$. How does one prove that the Yang-Mills equations $$ \frac{\partial}{\partial x^i}F_{ij}+[A_i,F_{ij}]=0 $$ from ...
10
votes
1answer
313 views

Solving Poisson's equation for $\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) \, \delta(y)$

Problem statement I took an exam, where I had the following task: Determine the electrostatic potential for the charge distribution $$\varrho(\mathbf{r}) = \sigma \cos\left(\frac{2 \pi}{L} x\right) ...
10
votes
2answers
370 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 ...