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 7294
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
Accepted
Hardy inequality
You can estimate
$\||x|^{-1}u\|_{L^2}\le C
\||x|^{-1}\|_{L^{n,\infty}} \|u\|_{L^{q,2}}$
in terms of the Lorentz norm $L^{q,2}$ with $q=\frac{2n}{n-2}$. This is slightly stronger than $L^q$, but not …
- 9,007
1
vote
A Inequality in the paper by Kenig, Ponce and Vega
Let me also mention the paper by D.Li, On Kato–Ponce and fractional Leibniz rule, which has quite complete results.
- 9,007
5
votes
Accepted
Characterization of locality in Fourier multiplier
Let $a$ be the Fourier transform of $f(\xi)$, in general $a$ will be a distribution. Then the action of $f(D)$ on a function is a convolution with $a$, that is $f(D)u=a*u$. Now, $a$ is compactly suppo …
- 9,007
2
votes
Accepted
On the Fractional Laplace-Beltrami operator
This is false even for the euclidean fractional Laplacian. For a cheap proof, the Fourier transform of $(-\Delta)^{a/2}u$ is $|\xi|^a\widehat u(\xi)$ which is not smooth at 0. For a more solid proof, …
- 9,007
8
votes
Accepted
$L^p-L^q$ boundedness of this simple singular oscillatory integral operator
Let me consider instead the operator
$$Sf(x):=\int e^{-i x y} \frac{f(y)}{|x-y|^{\alpha}}dy$$
which has the same properties as $T$ since $Tf(x)=Sf(-x)$. Writing
$$e^{-i x y}=e^{i|x-y|^{2}/2}e^{-i|x|^{ …
- 9,007
4
votes
Strichartz estimates and Schrödinger equation with derivative
It is not clear to me how you plan to close your argument unless you assume that the coefficient $\Phi$ is sufficiently small. I think it is not easy to get rid of first order terms by a perturbative …
- 9,007
3
votes
Accepted
$H^s$ norm of non-integer power of functions
In Christ-Weinstein, JFA 100 (1991) 87-109 you can find the fractional chain rule (Proposition 3.1)
$$
\|F(u)\|_{\dot H^s_r}\le C
\|F'(u)\|_{L^p}\|u\|_{\dot H^s_q}
$$
where $s\in(0,1)$, $p,q,r\in(1,\i …
- 9,007
2
votes
Accepted
Decay of solutions to Schrodinger equation with local minimum in potential
There are several decay results in the 1D case, but probably they are not enough for you. Goldberg and Schlag (Comm. Math. Phys. 251 (2004) 157–178) proved pointwise decay of the $L^\infty$ norm as $t …
- 9,007
1
vote
Accepted
About the continuity of the integral on the boundary of a ball
I would say so. Denote your integral by $b_u(x)=\int_{|x-y|=r}u(y)dH^{n-1}$.
Approximate $u$ in $H^1$ with test functions $u_j$. The property is certainly true for $u_j$ thus it is enough to prove tha …
- 9,007
2
votes
Accepted
Estimate for an oscillatory integral of the first kind
Write $s=\eta t$ and note that $I(s,y)$ solves the 1D Schrödinger equation $iI_s-3I_{yy}=0$. Thus it satisfies the sharp estimate $|I(s,y)|\le c_0s^{-1/2}$ where $c_0$ is a multiple of $\int|I(0,y)|dy …
- 9,007
3
votes
Short time existence for fully nonlinear parabolic equations
Some years ago we had a similar difficulty in finding results on fully nonlinear parabolic problems. Maybe these topics fell out of fashion before anybody wrote a reasonably complete wrap-up of known …
- 9,007
8
votes
What is dispersive estimate?
The intutitive picture is the following: there is a certain amount of "mass", which at time $t=0$ you might imagine concentrated in a heap near a location $x=0$. This heap evolves with time according …
- 9,007
1
vote
Accepted
An inequality of spacetime Banach space for non-linear Schrodinger equation
It is sufficient to write
$$u^2\bar{u}-v^2\bar{v}=(u-v)u\bar{u}+(u-v)v\bar{u}+(\bar{u}-\bar{v})v^2$$
and then differentiate each term.
- 9,007
2
votes
Boundedness of Riesz potential on Hardy space
I suspect this is false (without additional restriction. Maybe compact support?). Indeed take $u(x)=|x|^{-1/2}$ which is in $L^{2,\infty}$, then $I_{1/2}u$ is a real constant times the convolution $u* …
- 9,007
1
vote
Difference between semilinear and fully nonlinear
In my experience, fully nonlinear refers to an equation of the form $F(x,Du,D^2u)=0$ such that the linearization at $u=u_0$ is a linear equation of the required type (elliptic, hyperbolic etc.). In th …
- 9,007