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Results tagged with ap.analysis-of-pdes
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user 21907
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
5
votes
Minimal assumptions for existence of solutions of First order PDE
There are two a priori different points of view. The first one is the Lagrangean where you look at the ODE
$$
\dot x(t)=X(x(t)),
$$
where $X$ is your given vector field. Then, as you mention, a local …
- 16.2k
2
votes
On elliptic operators on non-compact manifolds
Too long for a comment.
I would say that your problem is a semi-global solvability question. You will find in chapter 26 of Hörmander’s ALPDO a precise definition for that property which suits well th …
- 16.2k
2
votes
Accepted
Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>2 $
Your functions $u_j$ are solutions of a semi-linear elliptic equation and, for $p>2$ the function
$t\mapsto\vert t\vert^{p-1}t=\phi(t)$ is $C^2$; as a consequence, each $u_j$ is $C^\alpha$ with some p …
- 16.2k
3
votes
How to understand the unique continuation result
If I understand things correctly, $u$ is vanishing on some non-empty open subset. Moreover, you have pointwisely on $\mathbb R^3$ the differential inequality
$$
\lvert \Delta u\rvert\le C\lvert u\rve …
- 12.9k
2
votes
$L^\infty$ estimate for elliptic PDE with mixed boundary conditions
Too long for comment. take for instance $f=0, g=0$. Then the mapping $h\mapsto u$ is a pseudo-differential operator which will have some Sobolev continuity properties for spaces $W^{s,p}$ with $p\in …
- 16.2k
1
vote
A variant of Hardy's inequality for "convolutions"?
I believe that for $n=3$ the optimal Hardy inequality is
$$
\int_{\mathbb R^3}\vert(\nabla w)(y)\vert^2 dy\ge \frac14
\int_{\mathbb R^3}\frac{\vert w(y)\vert^2}{\vert y\vert^2} dy,
$$
say for $w$ in t …
- 16.2k
8
votes
PDEs and algebraic varieties
A most important result is missing in the previous answers, namely the characterization by Lars Hörmander of hypoellipticity in his seminal paper,
On the theory of general partial differential operato …
- 16.2k
3
votes
Characterization of locality in Fourier multiplier
The only local pseudo-differential operators are the differential operators and this entails that the only local Fourier multipliers are polynomials. It is a classical result due to J. Peetre, Math. S …
- 16.2k
1
vote
Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
Too long for a comment. I guess that your question is "If I know that $\lvert D_x\rvert^{-\alpha} f$ belongs to $\mathrm{BMO}$, what can I say about the regularity of $f$?" I guess that $\alpha$ belon …
- 12.9k
1
vote
Symbol estimates using metric on the phase space
You are given a metric on the phase space, i.e. at each point $(x,\xi)$ in $\mathbb R^{2n}$ a positive definite quadratic form $g_{x,\xi}$ on $\mathbb R^{2n}$. You will use that metric (and the affine …
- 16.2k
3
votes
Singular support: equivalent definition
Let $U$ be an open subset of $\mathbb R^d$ and let $u\in \mathscr D'(U)$. Then we have
$$
(\text{supp } u)^c=\{x\in U, \exists V \text{open neighborhood of $x$ such that}\ u_{\vert V}=0\},
\tag{1}$$
…
- 16.2k
2
votes
Accepted
How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\...
Let $E$ be a fondamental solution of $\partial_{x_1}\square$. Then you have for $u$ compactly supported
$$
u=u\ast \delta=u\ast (\partial_{x_1}\square E)= (\partial_{x_1}\square u)\ast E,
$$
so that
$ …
- 16.2k
1
vote
Approximation of Hölder functions by Fourier series
Hölder functions periodic with period 1 satisfy the Dini criterion
$$
\int_0^{1/2}\frac{\vert u(x+t)- u(x)\vert}{t} dt<+\infty,
$$
thus their Fourier series are uniformly convergent (towards $u$). On …
- 16.2k
1
vote
Is the Fourier multiplier $\mathcal F(G(-\hbar^2 \Delta)\psi)(p) = G(|p|^2)\hat \psi(p)$ jus...
To make sense of your Fourier multiplier, you need only to assume that $p\mapsto G(\vert p\vert^2)$ is a temperate distribution on $\mathbb R^d$. It is true whenever $G$ is a continuous function incre …
- 16.2k
4
votes
Accepted
Vorticity equation for incompressible 2D fluid dynamics
The vorticity equation for the Euler equation in 3D is, with $\omega=\text{curl } v$,
$$
\dot\omega + (v\cdot\nabla)\omega-(\omega\cdot\nabla)v=0,
$$
so that if $v$ is two-dimensional, i.e.
$
v=\begin …
- 16.2k