All Questions
Tagged with operator-theory partial-differential-equations
290
questions
1
vote
0
answers
36
views
Principal Symbol of the Fractional Laplacian on Manifolds
In the euclidean space $\mathbb{R}^n$, we can define the Fractional Laplacian as
$$(-\Delta)^s f := \int |\xi|^{2s}\hat{f}(\xi)e^{ix\cdot\xi}d\xi.$$
The principal symbol is clearly $p(x,\xi)=|\xi|^{2s}...
- 323
1
vote
0
answers
48
views
deriving differential equation from difference of PDE solutions
Consider defining $$f(x,t)=g(x,t)-h(x,t)$$ where the PDEs describing $g(x,t)$ and $h(x,t)$ are $\it{known}$ but the solutions themselves are $\it{unknown}$. It could be impractical to solve the PDEs ...
1
vote
0
answers
15
views
What is the Propagator associated to a homogeneous Cauchy problem?
I came across this problem:
Consider the following Cauchy problem on $\mathbb{R}_t\times\mathbb{R}_x^n$
$$\begin{cases}\partial_tu+\omega\cdot\nabla_xu=f(t,x),\\ u|_{t=0}=u_0(x)\end{cases}$$
where $\...
- 598
0
votes
0
answers
48
views
Sectoriality of the 1-dimensional Laplace operator in $H^{-1}$
I know how to prove that the 1-dimensional Laplace operator is sectorial in $L^{2}$ with the domain being $H_{0}^{1}\cap H^{2}$. I've been wondering how to prove that it is also sectorial in $H^{-1}$ ...
1
vote
1
answer
47
views
Calculating the fractional Laplacian of a constant function
I have a question about calculating the Spectral fractional Laplacian of a constant.
Let us recall the definition of the Spectral fractional Laplacian for $0<s<1$:
$$
\left(-\Delta_{\text {Spec }...
- 123
2
votes
1
answer
94
views
Do exponentials in semigroups only have formal meaning?
Let $L$ be a linear differential operator and consider the PDE
$$\begin{cases} u_t + Lu = 0, \quad x \in \mathbb{R}^n\\
u = f, \quad t = 0\end{cases} \tag{1}$$
It is known that we may construct a ...
- 6,277
0
votes
0
answers
23
views
How to show that $A_h$ $\gamma$-converges to $A$ iff $\lim_{n\rightarrow \infty} \sup_{|f|_{L^2}\leq 1}\|R_h(f)-R(f)\| = 0. $
I am reading a paper by Butazzo, Dal Maso. Let $\Omega$ be a bounded open set in $\mathcal{R}^n$. Let $A_h$, $A$ $\subset \Omega$ be (quasi-)open. Consider the following two PDEs with Dirichlet ...
1
vote
0
answers
246
views
How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, I heard that $\operatorname{Id}-K$ is a proper map. Proper map ...
- 11
2
votes
1
answer
46
views
Question on the Gateaux derivative (being the principal part of a system of PDEs)
Let me start with a disclaimer. This is the first time ever dealing with Gateaux derivative, hence my knowledge around this topic is poor.
As I was going through this paper, I got stuck in the ...
- 2,964
3
votes
5
answers
260
views
How to give a closed form to $e^{a(x) \frac{d}{dx} + b(x)I}[f]$ in physicists style abuse of notation?
In Quantum Mechanics we have the famous time evolution result (here $a$ is a constant)
$$ e^{a \frac{d}{dx}}[f] = f(x+a) $$
Which is an abuse of notation but makes sense due to Taylor's Theorem.
In ...
- 17.5k
0
votes
0
answers
48
views
Question about equivalence between these two Ginzburg-Landau equations
I'm going through this paper and I'm stuck at showing the equivalence between the follwoing two equations:
where the first equation for $\alpha$ is the respective (1.4). In particular, I can't prove ...
- 2,964
0
votes
0
answers
38
views
Existence of a resonance (or lack of it) for Schrodinger operators
In dimension 1, consider two potentials $V_1,V_2\in C^\infty(\mathbb{R})$, both of them decaying exponentially fast at $\pm\infty$. Suppose that the Schrodinger operator $$
L_1=-\partial_x^2+V_1,
$$
...
- 151
1
vote
1
answer
56
views
Adjointiness, Poisson bracket and KdV equation
Consider this (I am only having issues with the proof, the example is added as context and definitions)
Example (Showing that KdV equation is bi-Hamiltonian) We can write the KdV equation in ...
- 2,302
3
votes
1
answer
185
views
Semigroup properties of spectral fractional Laplacian
I need to check if the different fractional Laplacians have the same properties regarding the semigroups they generate. All I know now is in the case of the Restricted Fractional Laplacian:
$(-\Delta)^...
- 123
1
vote
1
answer
70
views
Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian
I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...
- 159