Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with ap.analysis-of-pdes
Search options answers only
not deleted
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.
0
votes
Dirichlet problem for mean curvature equation
In $\mathbb{R}^3$, the Plateau problem for constant mean curvature surfaces was solved in
Hildebrandt, S., On the Plateau problem for surfaces of constant mean curvature, Commun. Pure Appl. Math. 23, …
- 39.1k
5
votes
Huygens' principle or finite speed of propagation?
Two points:
1.
In the wave equations community (in contrast to the community for more general hyperbolic systems) it is rather common to sloppily interchange "Huygen's Principle" with "finite speed of …
- 39.1k
4
votes
Accepted
Tangential Sobolev spaces
The assumption that $\Omega$ is bounded is in fact required. (So your attempt is the correct proof, once you fix the omission in the statement.)
Counterexample: let $\Omega$ be the upper half plane. L …
- 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 …
- 6,377
1
vote
Accepted
Spatially localised solution to the Schrödinger equation with potential is a combination of ...
The statement Terry alluded to is a consequence of the RAGE Theorem.
For a statement and proof of the (abstract) RAGE Theorem, see Section 5.2 of the 2nd edition of Teschl's Mathematical Methods in Qu …
- 39.1k
6
votes
Accepted
Quasilinear wave equations without (weak) null conditions and conjectures
What you've found is basically "survivor bias", so it helps for me to describe a bit where the null conditions came about.
Assertion 1: Quasilinear partial differential equations, in general, are too …
- 39.1k
3
votes
Accepted
A question on the proof of unique continuation for the case $u\in H^{2}$ in Le Rousseau, Leb...
I think one can answer your question (specifically the bold parts of your text) without having any knowledge of the precise argument of Lebeau; if you find the answer lacking, please let me know and a …
- 39.1k
2
votes
Accepted
Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta...
Let me address this comment.
Let $\phi \in C^\infty_c(\mathbb{R})$, then the function
$$ u(t,x) = \int_{\mathbb{R}} \exp(-i\xi^4 t + ix\xi) \phi(\xi) ~\mathrm{d}\xi$$
solve the first equation with ini …
- 39.1k
3
votes
Accepted
Does local gradient collinearity imply factorization
By continuity we may assume near $(0,0)$ neither $\nabla f$ and $\nabla g$ vanish. Let $u = \nabla f(0,0)$, extended as a constant vector field on $\mathbb{R}^2$. Let $v$ be a non-vanishing vector fie …
- 39.1k
2
votes
Accepted
Definitions of weak solutions for quasilinear wave equations
For simplicity I'll assume $u$ is scalar valued, but I am pretty sure the discussion also works for $u$ that is a section of some vector bundle over $M$ (if the wave operator is quasidiagonal). Additi …
- 39.1k
1
vote
An expansion for 2d Euler equation
Let's consider the simplified equation (which is really WLOG, since you are just absorbing constants)
$$ x \sqrt{-\ln x} = y $$
for $x,y > 0$ and $x,y \ll 1$. (Here $x$ is playing the role of $s_\epsi …
- 39.1k
6
votes
Accepted
Uniqueness of constructed solutions to the Helmholtz equation
The old Sherlock Holmes adage
When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
applies here. Since nothing else you did was wrong, it must be your id …
- 1,422
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
3
votes
Accepted
A simple bilinear estimate
Since $f$ is only defined for $x\in [0,1]$ and I can perform a change of variables to get $y \mapsto 1-y$, I think you are asking about the inequality
$$ \int_{[0,1]\times [0,\delta]} \frac{f(x)g(y)}{ …
- 39.1k
3
votes
Accepted
A Inequality in the paper by Kenig, Ponce and Vega
The inequality you cited is sometimes called a "fractional Leibniz rule", and is related to the Coiffman-Meyer theorem. An extension to all $\alpha \geq 0$ is available in
Fujiwara, Kazumasa; Georgiev …
- 39.1k