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Results tagged with ap.analysis-of-pdes
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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.
6
votes
5
votes
Accepted
Short-time Existence/Uniqueness for Non-linear Schrodinger with Loss of Several Derivatives
Just a few thoughts. The answer to your question depends on a number of key factors. To focus, let us consider a nonlinear term like $F(D_x^k u)$ and let us work in one space dimension.
1) How large …
- 9,007
2
votes
A hyperbolic PDE with infinite boundary conditions
Just to make sense of your problem, think of the following special case: $a=1$, $b=0$ are both constant functions. Then every solution can be written as $z(x,y)=p(x+y)+q(x-y)$, and you have $$ z_x(x,y …
- 9,007
2
votes
Accepted
Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$
Lipschitz continuous functions on $\Gamma$ are dense in $L^2(\Gamma)$ and are contained in $H^{1/2}(\Gamma)$.
- 9,007
4
votes
Strichartz estimates and Schrödinger equation with derivative
It is not clear to me how you plan to close your argument unless you assume that the coefficient $\Phi$ is sufficiently small. I think it is not easy to get rid of first order terms by a perturbative …
- 9,007
6
votes
Accepted
A question about the proofs of the Sobolev embedding theorem.
The classical Gagliardo-Nirenberg proof covers the limiting cases
$$\|f\|_{L^{\frac{n}{n-1}}}\le C\|\nabla f\| _{L^1}$$
and, for $f$ vanishing at infinity,
$$\|f\| _{L^\infty}\le C\|f\| _{W^{n,1}}$$
w …
- 9,007
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
3
votes
Accepted
Strichartz Estimates for radial Klein-Gordon equation
You can find a complete set of Strichartz estimates for the nonradial KG equation in Lemma 3 in this paper by Machihara, Nakanishi and Ozawa. Let me add that, usually, in proofs of local or global wel …
- 9,007
1
vote
The dependence of constant in a trace theorem on the diameter of domain
Since $Tu=T(\chi u)$ for any smooth cutoff $\chi$ equal to 1 near the boundary, the norm of the trace operator should be independent of the size of the open set (maybe it could be influenced by the do …
- 9,007
2
votes
Existence of solution to quasilinear parabolic PDEs
Some years ago we wrote a paper on degenerate parabolic equations (and, actually, systems) which you might find of help. We worked in Sobolev classes there, but I think you can adapt the techniques th …
- 9,007
2
votes
Accepted
Besov Characterization of Strichartz Estimate.
Look at Figure 3 in the paper by Ginibre-Velo, point $C_2$ is the estimate of $\|u\|_{L^4\dot B^{1/4}_{\infty,2}}$ in terms of the $L^1L^2$ norm of $F$, and one has $\dot B^{1/4}_{\infty,2}\hookrighta …
- 9,007
2
votes
Null sets in PDE
The meaning of your question is not completely clear to me. Anyway, the following remark might be useful: given a weak solution $u(t)$, the set of test functions $v(t)$ satisfying the identity against …
- 9,007
8
votes
Accepted
Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
If $u_j$ is such a basis, from $(\nabla u_j,\nabla u_k)=0$ for all $j\not=k$ it follows $(\Delta u_j,u_k)=0$ which means $\Delta u_j$ is a multiple of $u_j$, and unfortunately $\Delta$ has no eigenval …
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7
votes
Solitary waves and their symmetries
Palais' fantastic summary tells most of what is necessary to know if you want to understand solitons (and I re-read that paper with pleasure). I thought I'd add just one more perspective since the mor …
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2
votes
Seminorms in sharp Garding's inequality
Probably the sharpest result on this is due to Tataru, he proved sharp Garding for $C^2$ symbols (and even Fefferman-Phong for $C^4$ symbols). The paper is available here.
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