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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
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
1
vote
The integrability of fundamental solution of laplace equation follows from integrability of f ?
Let $E$ be the fundamental solution of the Laplace equation in $\mathbb R^n$ for $n\ge 3$: $E=c_n\vert x\vert^{2-n}$ so that $E$ is $L^1_{loc}$. For $f$ compactly supported, $\chi\in C^\infty_c$,
$$
…
- 16.2k
5
votes
Accepted
Quasi-linear System of First Order P.D.E.s of "Mixed" type
Let me change your notations. You deal with a 1D quasilinear system with size $N=4$: the standard Cauchy problem is
$$
\frac{\partial u}{\partial t}+A(t,x,u)\frac{\partial u}{\partial x}= f(t,x),\quad …
2
votes
Nonharmonic solutions of Laplace's equation
If $U$ is an open subset of $\mathbb R^n$, $f$ is a distribution on $U$ such that $\Delta f$ is analytic on $U$, then $f$ is analytic on $U$. This hypoellipticity result is true for any elliptic diffe …
- 16.2k
3
votes
$C^{2}$ estimates for elliptic equations
The parametrix $E$ of a second order elliptic operator with smooth coefficients is a singular integral (or pseudodifferential operator of order -2) and sends
$$
E:W^{s,p}\longrightarrow W^{s+2,p},\qua …
- 16.2k
5
votes
Accepted
Extension of solutions of PDEs with constant coefficients
If that property is satisfied, then "hypoelliptic analyticity" holds, which means that $\mathcal L f$ analytic implies $f$ analytic. For constant coefficient operators that property is equivalent to e …
- 16.2k
3
votes
unique continuation property for overdetermined elliptic PDE
(1) The most classical result, due to Aronszajn, and later to Calder\'{o}n and (independently) to H\"ormander. Take a second order elliptic operator $P$ with Lipschitz and real coefficients in the pri …
- 16.2k
2
votes
Accepted
What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are...
Change the variables: define
$$
u(t,s,x)=v(\underbrace{\frac{t+s}{2}}_{\tau},\underbrace{{t-s}}_{\sigma},x).
$$
You get
$
\partial_t u+\partial_s u=\partial_{\tau} v
$
and the equation becomes
$$
\par …
- 16.2k
3
votes
Analyticity of the solutions of PDE
Take the Lewy operator $\mathcal L$ in $\mathbb R^3_{x,y,t}$
$$
\mathcal L=\partial_x+i\partial_y+i(x+iy)\partial_t.
$$
There exists a set $S$ of second category in $C^\infty(\mathbb R^3)$ such that f …
- 16.2k
3
votes
Decay rate of nonlocal differential operator?
You have an explicit formula: the symbol of $\chi_jm(D)\chi_k$ is
$$
a_{jk}(x,\xi)=\chi_j(x)\iint e^{-2\pi iy\eta}m(\xi+\eta)\chi_k(y+x) dyd\eta.\tag C
$$
It is difficult to handle this with $\chi_j$ …
- 16.2k
3
votes
Accepted
Interpretation of a parameter in forming a pseudodifferential operator
Let me answer to your second query and make $h=1$. You have
$$
Op_1(a(x) \xi)= a(x) D_x,\quad \text{with $D_x=-i\partial_x$},
$$
$$
Op_0(a(x) \xi)= D_x a(x),
$$
$$
Op_{1/2}(a(x) \xi)= \frac 12D_x a(x) …
- 16.2k
2
votes
Accepted
Weak divergence implies weak differentiability of components?
So $\sigma=\sum_{1\le j\le n}\sigma_j(x)\frac{\partial}{\partial x_j}$ is a vector field with distributions coefficients $\sigma_j$ and divergence in $L^2$:
$$
\sum_{1\le j\le n}\frac{\partial \sigma …
- 16.2k
2
votes
Solvability for constant-coefficient partial differential operators
For your constant coefficient operator $L(D)$, you want a fundamental solution $G$ such that $\hat G$ is a multiplier of $\mathcal S'$. This is not even true for the Laplace equation: the fundamental …
- 16.2k
1
vote
Does these commutator estimates bound in $L^{2}$
Take a pseudodifferential operator with a symbol $p\in S^1_{1,0}$ and $f$ a Lipschitz-continuous function. Then the commutator
$$
[p(x,D),f]
$$
is bounded on $L^2$. When $f$ is $C^\infty$ with bounded …
- 16.2k
1
vote
Proper sobolev spaces invariant under no-linearities
Let $f:\mathbb C\rightarrow\mathbb C$ be a $C^\infty$ function such that
$f(0)=0$.
Let $1\le n\in \mathbb N$ and $s>n/2$. Then
$$\forall u\in H^s(\mathbb R^n),\quad f(u)\in H^s(\mathbb R^n).
$$
In pa …
- 16.2k