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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.
41
votes
Motivation for and history of pseudo-differential operators
Let us follow the history of classical PDE with a few (hopefully) well-chosen examples illustrating the rôle of pseudodifferential operators in classical analysis.
(1) Before 1950. Prehistory. A lon …
- 16.2k
28
votes
Does Physics need non-analytic smooth functions?
A strong argument is given above on the heat equation; let me be more specific. The heat equation, one of the most basic in PDE and mathematical physics, already known to Fourier, is
$$
L=\frac{\parti …
- 16.2k
8
votes
Is there any way to rewrite a partial differential equation using language of differential f...
Let $\mathcal M$ be a smooth manifold. A linear PDE on $\mathcal M$ is a sum of terms
$$
X_1\dots X_N u,\quad\text{where the $X_j$ are smooth vector fields.}
$$
We may use the convention that if $N=0$ …
- 16.2k
8
votes
what's the motivation of Weyl calculus ?
It is true that the initial motivation for Hermann Weyl in 1926 was linked to quantum mechanics and his convention was indeed ensuring that real-valued Hamiltonians get quantized by (formally) selfadj …
- 16.2k
8
votes
PDEs and algebraic varieties
A most important result is missing in the previous answers, namely the characterization by Lars Hörmander of hypoellipticity in his seminal paper,
On the theory of general partial differential operato …
- 16.2k
7
votes
when a pseudo-differential operators to be compact?
First a simple remark: in the formulation of the question $\mu$ should be replaced by $\mu/2$ to get
$S(m,g)=S^\mu_{1,0}$.
Next the "if and only if" is correct but misleading since it is a conditio …
- 16.2k
7
votes
Accepted
Heat Equation on $[0,T] \times \mathbb{R}^n$
Fourier transform in $x$ gets you there:
$
\dot v+\eta\vert\xi\vert^2 v=g(t,\xi),\quad v(0)=v_0,
$
so that
$$
v(t,\xi)=e^{-t\eta \vert\xi\vert^2} v_0(\xi)+\int_0^te^{-(t-s)\eta \vert\xi\vert^2} g(s,\x …
- 16.2k
6
votes
Accepted
Hormander's bracket condition for the adjoint of an operator
The hypoellipticity result is more precise:
you have
$$
Lu \in H^s_{loc}\Longrightarrow u\in H^{s+2-\delta}_{loc}\quad\text{ for some $\delta\in [0,2)$,}
$$
and that $\delta$ is linked to the number o …
- 16.2k
6
votes
Accepted
When is the adjoint of a hypoelliptic operator also hypoelliptic?
Hormander's operator $L=X_0+\sum_{1\le j\le k} X_j^2$, where the $X_j$ are real smooth vector fields with the Lie algebra of $\{(X_j)\}_{0\le j\le k}$ generating the tangent space is hypoelliptic as …
- 16.2k
6
votes
Are Besov spaces $B^{s}_{p,q}$ invariant under Fourier transform?
Let me note $\phi_k(D)$ the Fourier multiplier $\phi_k(\xi)$, i.e.
$
\text{Fourier}\bigl(\phi_k(D)u\bigr)(\xi)=\phi_k(\xi)\hat u(\xi).
$
$\bullet$ The answer to (1) is yes since
$$
\Vert{u}\Vert_{L^1 …
- 16.2k
5
votes
Easy Garding Inequality
I understand that you are dealing with semi-classical symbols
$$
b(x,\xi, h)=a(x,h\xi), \quad \vert\partial_x^\alpha\partial_\xi^\beta b\vert\le
C_{\alpha\beta} h^{\vert \beta\vert} m(x),\quad 0<h\le …
- 16.2k
5
votes
Accepted
Extension of solutions of PDEs with constant coefficients
If that property is satisfied, then "hypoelliptic analyticity" holds, which means that $\mathcal L f$ analytic implies $f$ analytic. For constant coefficient operators that property is equivalent to e …
- 16.2k
5
votes
explicit solution of fractional laplacian in R^N
The operator $(-\Delta)^s$ in $\mathbb R^N$
is the Fourier multiplier $c_{s,N}\vert \xi\vert^{2s}$, so the Fourier transform of the fundamental solution $E_{s,N}$ should be homogeneous with degree $-2 …
- 16.2k
5
votes
Accepted
Quasi-linear System of First Order P.D.E.s of "Mixed" type
Let me change your notations. You deal with a 1D quasilinear system with size $N=4$: the standard Cauchy problem is
$$
\frac{\partial u}{\partial t}+A(t,x,u)\frac{\partial u}{\partial x}= f(t,x),\quad …
5
votes
Example for the Sobolev embedding theorem when p=n.
Take $u$ in $\mathscr S'(\mathbb R^2)$ with
$$
\hat u(\xi)=\frac{\mathbf 1(\vert\xi\vert\ge 2)}{\vert\xi\vert^3 \ln\vert\xi\vert},\quad
\vert \xi\vert^2\hat u(\xi)=\frac{\mathbf 1(\vert\xi\vert\ge 2)} …
- 16.2k