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Results tagged with ap.analysis-of-pdes
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user 3948
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
31
votes
Accepted
Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and do...
In regards to the question of the "consensus" or "correctness", I will only point out that Tristan Buckmaster has had a proven record of studying nonuniqueness problems for low-regularity solutions in …
- 39.1k
28
votes
Accepted
Why don't we study hyperbolic equations as elliptic and parabolic equations?
Why we do not study such estimates for hyperbolic equations?
Because they are false.
Now: you may ask "why are they false?" This is a fairly deep question, and answers often involve discussion of p …
- 39.1k
22
votes
Accepted
Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes?
Okay, after figuring out which paper you were trying to link to in the third link, I decided that it is better to just give an answer rather then a bunch of comments. So... there are several issues at …
- 39.1k
19
votes
Method of characteristics for higher order PDEs in more than two variables
I hope to use this answer to convince you that in general the method of characteristics cannot work for higher order PDEs in more than 2 variables. Nevertheless, there are some ideas in PDEs that are, …
- 39.1k
19
votes
Accepted
Einstein field equations in perspectives from PDE and functional analysis
The statement
It seems that the classical programme of the PDE community, i.e., (i) existence (ii) uniqueness (iii) regularity, heavily employing concepts from functional analysis, has not found prom …
- 39.1k
17
votes
Can an integral equation always be rewritten as a differential equation?
In general, no. An integral equation can be non-local, whereas a differential equation is local (in the sense that it can be described by a function over the jet-bundle). As an illustration
Let $K(x) …
- 39.1k
16
votes
When to use more exciting function spaces than ordinary Sobolev spaces?
In many aspects of dispersive PDEs, the "optimal" function spaces are those adapted to the symbol of the linear evolution. They were introduced by Bourgain for the nonlinear Schroedinger equation (and …
- 39.1k
13
votes
What are the interesting cases of the generalized Korteweg-de Vries equation?
José is correct in his comment. Just to elaborate: in the linear case, one can easily study the equation using Fourier methods. Let $\tilde{u}$ denote the space-time Fourier transform and $\hat{u}$ de …
- 39.1k
11
votes
Accepted
Does the Poincaré inequality hold on annular domains?
I will prove the stronger result without the subtraction of $\bar{f}$. As we know $\int |f|^2 = \int |f - \bar{f}|^2 + \int |\bar{f}|^2$, the result without subtracting $\bar{f}$ would imply what you …
- 39.1k
11
votes
Accepted
leray schauder fixed point and schauder fixed point
Note that Leray-Schauder is usually proven by using the hypotheses to construct a mapping that satisfies the conditions of the Schauder fixed point theorem, and then appealing to the Schauder fixed p …
- 39.1k
11
votes
Accepted
Does harmonic map heat flow of a curve always fully converge to a geodesic?
The situation is actually quite complicated, it seems.
In the case where the target manifold is real analytic, Leon Simon's results in Asymptotics for a Class of Non-Linear Evolution Equations, with …
- 39.1k
10
votes
Where do some "energy identities" in PDE theory come from?
First a comment: in the context of nonlinear wave and Klein-Gordon equations, the venerable "ABC method" of Cathleen Morawetz is literally "mucking around until you see something". (The A, B, and C re …
- 39.1k
9
votes
Accepted
Do we have Pohozaev's identity on compact manifolds without boundary？
To answer this question, it is better to understand Pohozaev's identity using the heuristic argument given in
Berestycki, Henri; Lions, Pierre-Louis, Nonlinear scalar field equations. I: Existence of …
- 39.1k
8
votes
PDEs as a tool in other domains in mathematics
Another not-quite-yet connection which I learned from Lax's Hyperbolic PDE book: one can, technically speaking, extract the Riemann hypothesis from the scattering rates of certain "automorphic waves". …
8
votes
Accepted
Looking for references to study $U^p$ and $V^p$ spaces
You can take a look at Herbert Koch's contribution in
Koch, Herbert; Tataru, Daniel; Vişan, Monica, Dispersive equations and nonlinear waves. Generalized Korteweg-de Vries, nonlinear Schrödinger, wave …
- 39.1k