All Questions
1,304 questions
2
votes
1
answer
336
views
Is this a "contradiction" on stochastic Burgers' equation? How to understand it?
For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...
10
votes
1
answer
974
views
$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$
I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
- 261
7
votes
2
answers
573
views
Existence of spectral gap
I would like to start by saying that any comment or idea is highly appreciated.
Let us observe that for Hilbert-Schmidt operators $H_1,H_2$ on an infinite-dimensional separable complex Hilbert space $...
- 95
11
votes
2
answers
1k
views
Concentration compactness. Can this concept be stated in a theorem?
I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.
When I approached the speaker ...
- 501
2
votes
0
answers
93
views
How to use Fredholm alternative to check that there are only finite eigenvalues of $H$ on the imaginary axis?
On $\mathbb{R}^3$, we consider the operator
\begin{equation}
\mathcal{H}= \left( \begin{matrix}
-\Delta +1 -2 \phi^2 & -\phi^2 \\
\phi^2 & \Delta -1 +2 \phi^2
\end{matrix} \right) , D(...
- 429
2
votes
0
answers
126
views
Mixed partial derivatives of planar functions converging to delta distribution
Given a sequence $(f_k)_{k\in\mathbb{N}}\subset C^2(\mathbb{R}^2)$ of strictly positive functions $f_k\equiv f_k(x,y)$ with $\|f_k(x,\cdot)\|_{L^1}=1$ for all $x\in\mathbb{R}$ and such that for each $...
- 51
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}
\...
- 287
6
votes
2
answers
519
views
Existence of an integral equation (Faedo-Galerkin, Banach fixed point, Picard-Lindelof)
This question is concerning the paper, particularly the proof of Lemma 2.1 in Section 2.1:
Matas, A., Merker, J. Existence of weak solutions to doubly degenerate diffusion equations, Appl Math 57 (...
- 266
0
votes
0
answers
104
views
Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 61
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 ...
- 943
1
vote
2
answers
536
views
Non-closed range space of Laplace operators?
Set $ -\Delta: H^2(\mathbb{R}^3) \subseteq L^2(\mathbb{R}^3) \to L^2(\mathbb{R}^3) $. Then $ \mathcal{R}(-\Delta) $ is non-closed?
Sorry if this question is trivial. I am not familiar with theory of ...
- 269
8
votes
0
answers
251
views
Smoothness of solution map for PDE
I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
- 2,247
2
votes
1
answer
146
views
On $s$-harmonic functions
Is this statement true?
A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. ...
- 407
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
3
votes
0
answers
109
views
Examples/applications of parabolic PDEs that are not posed on domains or manifolds
Are there any examples of parabolic PDEs
$$u' - Au = f$$
posed in a Gelfand triple setting $V \subset H \subset V^*$ with $V$ and $H$ chosen NOT as spaces of functions (or distributions) over a domain ...
2
votes
1
answer
997
views
Derivative and Jacobian determinant of solution of ODE [closed]
Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
...
user124345
2
votes
1
answer
178
views
References for Neumann eigenfunctions
I am looking for references on eigenfunctions with Neumann boundary condition.
In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...
- 721
1
vote
0
answers
81
views
Compact imbedding for weight space
We begin with some definitions. Let $\gamma \geqslant 1,\,p \in \left[ {1,\infty } \right)$, we define
$$L_\gamma ^p\left( {0,1} \right) = \left\{ {v:\left( {0,1} \right) \to \mathbb{R}:{{\left\| v \...
- 183
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 $S_{r,s}^{\theta}$: $\...
- 11
9
votes
3
answers
2k
views
Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407
7
votes
2
answers
641
views
Decay of solutions to Schrodinger equation with local minimum in potential
Consider the one-dimensional Schrodinger operator on the real line $\mathbb{R}$ given by
$$ L = - \partial_x^2 + V $$
where $V$ is a potential with the following properties:
$V$ is non-negative, and ...
- 39.1k
5
votes
1
answer
361
views
Exponential decay of solution in $L^p$ with $p>2$
Consider the following evolution equation
$$u_t=\Delta u$$
in a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$, with smooth initial conditions $u_0\geq 0$ and homogeneous Dirichlet ...
- 161
2
votes
1
answer
339
views
Why is this test function admissible? [Paper explanation]
Reading Non-linear Elliptic and Parabolic Equations Involving Measure Data by Boccardo$\&$Gallouet , I had trouble understanding the following:
Why is $\psi(u_n)\chi_{(0,t)}$ admissible as a ...
- 227
4
votes
1
answer
336
views
Fundamental gap for Schrödinger operator
Consider $ \Omega$ a smooth bounded domain in $ \mathbb R^N$.
I am interested in the gap between the first and second eigenvalues of the operator $ -\Delta + V(x)$. Let $ \phi_1>0$ and $ \phi_2$ ...
- 1,385
1
vote
0
answers
88
views
Reference request: PDE of the form $(\Delta - |u|^2)f = F(u)$
I am interested in equations of the form
$$(\Delta -|u|^2)f = F(u)$$
where $F$ depends on $u$ and preferably on its derivative, too. $u$ is supposed to be given and $f$ the unknown. More precisely I ...
- 469
4
votes
2
answers
692
views
Uniqueness of viscosity solutions of Hamilton-Jacobi equation
Consider the following Hamilton-Jacobi (HJ) equation:
$$u_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n ...
user99249
5
votes
2
answers
2k
views
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $
I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \...
- 371
5
votes
0
answers
135
views
Relationship between continuous vector fields and divergence measure fields in dimension $\ge 2$
Let $\Omega \subset \mathbb R^d$ with $d \geq 2$ (I am mostly interested in the case when $\Omega$ is the unit ball). A vector field in $L^p(\Omega,\mathbb R^d)$ is called a divergence measure field ...
- 437
2
votes
0
answers
52
views
Reference Request: Dirichlet operators with singular coefficients
Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by
\begin{align*}
\mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
- 778
2
votes
1
answer
196
views
Support of functions in Fourier domain
Let $\mathcal F$ be the Fourier transform. I would like to understand whether being in a Sobolev space implies that the Fourier transform of a function is necessarily supported on a compact ball up to ...
- 23
3
votes
0
answers
127
views
Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?
My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider,
$\dot{x}=Ax$, where $x$ is the infinite dimensional ...
- 2,281
2
votes
0
answers
76
views
wave equation with L^2 boundary data via spectral decomposition
It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation
\begin{equation}\label{pf2}
\begin{aligned}
\begin{cases}
\partial^2_{t}u- \Delta u=0\,\...
- 4,115
1
vote
1
answer
468
views
"Combining" two differential equations into one
The Setup: Suppose $\Omega$ is a bounded, open, connected, simply connected subset of $\mathbb{R}^2$ with smooth boundary. Suppose that I am given a function $\Phi:\mathbb{R}^2\to\mathbb{R}$ and two ...
- 597
10
votes
1
answer
3k
views
Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user69109
4
votes
1
answer
267
views
Scaled Harnack inequality $\sup_{B_r} v \le c\,(1-r)^{-p}\, \inf_{B_r} v$
Where can I find a proof of the following scaled version of Harnack inequality?
Let $v$ be a non-negative solution of ${L}u = 0$ in $B_1$, with $L$ a uniformly elliptic operator. Then, for $r<1$,...
- 839
4
votes
0
answers
220
views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
- 5,380
2
votes
0
answers
71
views
Example of BV vector field $c$ without bounded divergence such that $u$ is bounded where $u_t + div(cu) = 0$
What is an example of vector field $c: \mathbb R_+ \times \mathbb R^N \to \mathbb R^N$ with $c \in L^1(\mathbb{R}_+, BV(\mathbb R^N))$ without bounded divergence $div_x c$ but such that there exists a ...
- 839
3
votes
0
answers
102
views
Determining what happens to the spectrum of Schrödinger operator as boundary condition changes
I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...
- 251
1
vote
1
answer
170
views
A time dependent variational problem coming from a second order linear PDE
Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$.
Consider the problem of finding $u:\Omega\times[0,T]\to\...
- 597
4
votes
0
answers
139
views
Smooth dependence of solution to elliptic pde depending on parameter
I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like
$$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is ...
- 1,385
5
votes
0
answers
214
views
Compactness of multiplication operators
Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
- 6,035
1
vote
0
answers
74
views
Fourier transform of a Sobolev function dependent on a "parameter"
Let $u\in\mathcal{S}(\mathbb{R}^n)$, let $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, such that
$$ V(x,0)=u(x),\quad V(x,\cdot)\in C^0([0,\infty)),\quad\forall x\in\mathbb{R}^n,$$
and ...
- 339
1
vote
1
answer
84
views
A completion equality in product of Sobolev spaces
Let $\Omega$ be a nice bounded domain of $\mathbb{R}^n$. For $s\ge 0$ we define
$$H_{0}^{s}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{H^{s}(\Omega)}.$$
My question: is the following equality true?
$$...
- 81
1
vote
1
answer
149
views
first order derivative of the parabolic equation
Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE
$$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, ...
- 1,399
3
votes
1
answer
300
views
Regularity and normal trace of "Hdiv" measures
In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...
- 5,380
12
votes
2
answers
2k
views
Reference on Minty's trick
I am searching for a precise reference for the following result:
Consider $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ a nondecreasing function.
Assume that a sequence of nonnegative functions $(u_n)_n$ ...
- 3,425
5
votes
1
answer
766
views
Weak convergence in $W^{1,p}_0$
Note from the answerer : this question stems from this article.
I ask this question in https://math.stackexchange.com/questions/1206617
I have a bounded sequence $(u_n)$ from $W^{1,p}_0(\Omega)$ so ...
- 277
3
votes
2
answers
849
views
Integral of fractional Laplacian is zero
Is it true that $$\int_{\mathbb{R}^N}(-\Delta)^su(x) dx = 0,$$
where $(-\Delta)^s$ is the fractional Laplacian?
user60665
2
votes
0
answers
71
views
Strict Riesz's rearrangement inequality when function is not nonnegative
The strict Riesz rearrangement inequality (Lieb- and Loss's book Analysis, Section 3, Theorem 3.9 ,page 93) says that if the functions $f,g,h,$ are all nonnegative and $g$ is strictly symmetric ...
- 379
2
votes
0
answers
163
views
Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,115