All Questions
1,304 questions
2
votes
0
answers
145
views
Integral estimate for the solution of the heat equation
Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality?
$$
\int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
CommunityBot
- 1
1
vote
1
answer
148
views
Understanding a family of Sobolev-type inequalities
I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:
Denote the following inequality as $S_{r,s}^{\theta}$: $\...
- 2,670
1
vote
0
answers
46
views
Decomposition of the space of Radon measures with respect fractional harmonic capacity?
It is well know that there is a generalization of Lebesgue decomposition theorem in the following way:
Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
- 6,367
0
votes
1
answer
233
views
Existence of subsequences convergence with weak topology
Let $\left\{ {{\varphi _n}} \right\}$ is the sequence bounded in ${L^\infty }\left( {0,\infty ;H_0^1\left( {0,1} \right)} \right)$. Is there exists $\varphi \in {L^\infty }\left( {0,\infty ;H_0^1\...
- 29.8k
3
votes
1
answer
752
views
A distributional normal derivative for functions in $H^1(\Omega)$
Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
- 1,724
5
votes
0
answers
214
views
Compactness of multiplication operators
Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
- 6,035
3
votes
1
answer
148
views
Smoothing-Strichartz estimates for the heat-Schrodinger evolution
Consider the heat-Schrodinger evolution $e^{(1+i)t\Delta}$, $t\geq 0$. For simplicity, suppose to work in dimension three. Due to the Strichartz estimates for the Schrodinger equation, and the ...
- 39.1k
1
vote
0
answers
169
views
A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
- 1,461
9
votes
1
answer
1k
views
Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
4
votes
0
answers
164
views
What's the essential definition of resonance of Schrodinger operator?
Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...
- 429
2
votes
1
answer
178
views
References for Neumann eigenfunctions
I am looking for references on eigenfunctions with Neumann boundary condition.
In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
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0
votes
0
answers
324
views
Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 63.9k
2
votes
0
answers
144
views
Is logarithmic convexity of the heat kernel with complex time a general fact?
Suppose $H$ is a non-negative self-adjoint operator acting on $L^2(\mathbb{R}^n)$, and generates an analytic semigroup with kernel satisfying a Gaussian upper bound, i.e., if we denote $K(t,x,y)$ the ...
- 63.9k
0
votes
1
answer
338
views
The part of an operator as an analytic generator
Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$.
Let $Y$ be another Banach space embedded in $X$. We consider$A_Y$, the part of $A$ in $Y$, defined as the ...
- 63.9k
2
votes
0
answers
149
views
Projection semigroup of an isolated eigenvalue
I'm currently working with a paper and I don't get something there. Let $A$ be a closed operator on a Banach space $X$ and $\lambda \in \sigma(A)$ an isolated eigenvalue, i.e. there is a $r > 0$ ...
- 63.9k
2
votes
0
answers
207
views
Smoothing properties of analytic semigroups
Assume $A$ is a second order operator, generator of a positive analytic semigroup on the $L^p$-spaces with $p\in (1,\infty)$ and domain $D(A_p)=W^{2,p}$. Do we have regularity estimates
$\|T_p(t)f\|_{...
- 63.9k
4
votes
1
answer
655
views
Generator of Laplace operator as analytic semigroup on $L^1(\mathbb{R}^n)$
The Laplace operator is the generator of an analytic semigroup on $L^p(\mathbb R^n)$
for $1 < p < \infty$. Is the same true for $L^1(\mathbb R^n)$? If it is, could someone give a reference? The ...
- 63.9k
1
vote
0
answers
353
views
Heat semigroup ultracontractive?
Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
- 63.9k
2
votes
1
answer
181
views
Estimate of semigroup with dual norm?
Consider a semigroup $(T(t))_{t\in\mathbb{R}^+}$ generated by a densely defined strictly positive symmetric linear operator $A: D(A) \subset X \to X$, where $X$ is a Banach space with norm $\|\cdot\|$....
- 63.9k
1
vote
2
answers
629
views
Regularity of the mild solution of a semilinear evolution equation
Let
$T>0$
$H$ be a separable $\mathbb R$-Hilbert space
$u_0\in H$
$(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ be an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;\;\;\text{for all }n\in\...
- 63.9k
1
vote
1
answer
430
views
Generator of an analytic semigroup of operators
I have an operator of the following form:
$$ A = \begin{bmatrix} 0 & h_1 & h_2 \\ 0 & \Delta& h_3 \\ 0 & 0 & h_4 \end{bmatrix} $$
which results from a coupled PDE-ODE system ...
- 63.9k
3
votes
1
answer
219
views
Positivity of generalised heat kernels
Let $K_\alpha(t,x)$ be the (generalised or fractional) heat kernel which corresponds to the fractional heat equation (I'm not sure that's the right name) in $\mathbb R^n$
$$
u_t=(-\Delta)^\alpha u, \...
- 63.9k
1
vote
1
answer
284
views
Reformulation of the classical Navier-Stokes equation as a semilinear evolution equation and corresponding mild solutions
Let
$d\in\mathbb N$
$\lambda^d$ denote the Lebesgue measure on $\mathbb R^d$
$\Lambda\subseteq\mathbb R^d$ be open
$\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\...
- 63.9k
2
votes
1
answer
197
views
Compactness of semigroups, boundary conditions
I have a question about compactness of semigroups and boundary conditions.
Let $\Omega$ be an unbounded domain of $\mathbb{R}^d$ with smooth boundary and $m(\Omega)=\infty$. Then we can define two ...
- 63.9k
3
votes
0
answers
367
views
Reference on semigroup theory and fractional heat equation
Consider the Dirichlet problem associated to the classical heat equation $\partial_t u - \Delta u = 0$ and to the fractional heat equation $\partial_t u + (- \Delta)^s u = 0$.
Where can I find a ...
- 63.9k
3
votes
1
answer
138
views
$C_0$ semigroups on parameterized Banach spaces or moving domains
Is there any literature corresponding to one or two-parameter semigroups such that e.g. $T(t) \in \mathcal{L}(X(t))$ or $T(s,t) \in \mathcal{L}(X(t),X(s))$ for parameterized Banach spaces $X(t)$?
I ...
- 63.9k
3
votes
1
answer
217
views
What is the meaning of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$?
The name of the operator $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\partial \overrightarrow{v}^2}]$ is one-parameter semigroup.
For example one writes $ EXP[\frac{ \theta }{2}\frac{\partial^2}{\...
- 63.9k
3
votes
1
answer
218
views
Short time $L^1$ bounds for semigroups obtained from elliptic operators
Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary, and let $L$ be a negative definite second order elliptic differential operator defined with $\mathcal{D}(L) \subset H^2(\Omega)$, given ...
- 63.9k
5
votes
1
answer
264
views
Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ generates?
Let $X$ be a Banach space (could work for over $\mathbb{R}$ as well?)
Let $A\colon D(A)\subset X\to X$ be a sectorial operator, and $e^{tA}$ be the semigroup generated by $A$.
It is well-known that ...
- 4,713
7
votes
3
answers
1k
views
Reference on semigroup theory and parabolic PDEs
Recently started to study semigroup theory. My background is equivalent to the first three chapters of the Jack Hale's book "Asymptotic behavior of dissipative systems".
Looking for a reference to an ...
- 63.9k
5
votes
1
answer
512
views
$C_0$-semigroups applications
My graduation thesis was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I would like to know if there is ...
CommunityBot
- 1
7
votes
1
answer
439
views
About the convergence rate for an approximation to the heat kernel
Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} \...
- 63.9k
1
vote
0
answers
136
views
Topology on function spaces for pointwise convergence
Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...
- 11
2
votes
0
answers
76
views
Weak Hessian for solutions to certain quasilinear elliptic PDEs
In Chapter 4 of their famous treatise Linear and quasilinear elliptic equations, Ladyzhenskaya and Uraltseva deal with equations of the form
$$\sum_{i=1}^n\frac{d}{dx_i}a_i(x,u,\nabla u)+a(x,u,\nabla ...
- 3,146
2
votes
1
answer
497
views
Some properties of fractional Dirichlet heat kernel
Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition：
\begin{equation}
\...
- 17.2k
3
votes
1
answer
441
views
Courant nodal domain theorem for fractional Laplacian
Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\...
- 17.2k
1
vote
0
answers
88
views
Explanation for the energy method used here
I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...
- 383
3
votes
1
answer
300
views
Regularity and normal trace of "Hdiv" measures
In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
- 16.5k
0
votes
0
answers
88
views
Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$
Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that
$$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...
- 183
1
vote
2
answers
424
views
Regular Lagrangian flow for the problem $\frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x))$
Consider the problem
$$(\star) \quad \begin{cases} \frac{d}{dt} X(t,x) = \chi_{\{x>0\}}(X(t,x)), &t \in [0,T],\\
X(0,x) = x, &x \in \mathbb R
\end{cases}
$$
where $\chi$ denotes the ...
- 60.6k
2
votes
0
answers
169
views
A basic question about the Spectral Theorem
Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...
- 434
5
votes
2
answers
2k
views
Elementary calculus estimate or not?
Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
- 4,899
1
vote
0
answers
166
views
Solution to Heat Equation By Projection
Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\...
- 5,405
4
votes
0
answers
170
views
Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$
Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
CommunityBot
- 1
1
vote
0
answers
80
views
One dimensional periodic travelling waves to some pde
Travelling wave equation on one dimension to Gross Pitaeavkii equation is $$ \phi '' +ic\phi'+\phi (1-|\phi|^2)=0\qquad (1) $$ where $c\in (0,\sqrt{2})$ and $ \phi$ is a complex valued function. I am ...
- 383
8
votes
1
answer
498
views
Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?
Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space:
$$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
- 3,574
1
vote
0
answers
45
views
How can we show that a solution depends on more than one variable? [closed]
I have obtained theoretically a solution to a nonlinear Schrödinger-type equation in dimension two. I also proved that is not constant. Now, I wonder if it depends on two variables and not only in one,...
- 383
2
votes
2
answers
361
views
Estimate of a solution of Schroedinger equation for a free particle
Let $\psi(x,t)$ be a solution of the Schroedinger on the line
$$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$
One assumes that $\psi(x,0)$ "behaves well" as $...
- 188k
1
vote
0
answers
110
views
Trace and second-order inverse trace on space with Gibbs measure
Consider $(t, x)\in [0,T]\times (\mathbb{R}^d,d\mu)$, where the measure $d\mu(x)=K^{-1}\exp(-U(x))dx$ is a reasonable Gibbs measure (it satisfies a Poincaré or log-Sobolev inequality. One can, for ...
- 63.9k
1
vote
0
answers
108
views
Extension of a compactly supported pseudo-differential operator
Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and $P \in \Psi^{m}(\Omega)$ a compactly supported pseudo-differential operator, that is, the kernel of $P$, is compact. Is it true that $P$ extends ...
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