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Uniform bound on the first moment for a perturbed advection-diffusion equation

I am studying the solution $u = u(t,x)$ to the following problem on the positive half-line: $$ \begin{cases} u_t = u_{xx} + u_x - \frac{1}{1+t}(xu)_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x = ...
Garou Garou's user avatar
3 votes
1 answer
86 views

$L^{1}$-convergence to steady states for an advection-diffusion equation on the half real line

I consider the following problem on the half real line $$ \begin{cases} u_t = u_{xx} + u_x, & \quad t > 0, \, x > 0, \\[2mm] -u_x|_{x=0} = u|_{x=0}, & \quad t > 0, \, x = 0, \\[2mm] u|...
Garou Garou's user avatar
3 votes
0 answers
203 views

On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
mathamphetamine's user avatar
2 votes
1 answer
237 views

Approximation of Hölder functions by Fourier series

Let $Q$ be a bounded domain in $\mathbf R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$. Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2\...
user500030's user avatar
2 votes
0 answers
104 views

Relations between different "propagation of chaos" type results?

My questions come from the paper Logarithmic Sobolev inequalities for some nonlinear PDE’s written by F. Malrieu (May 2001). The basic set-up is that we have a $N$-particle system $(X^{i,N}_t)_{1\leq ...
Fei Cao's user avatar
  • 730
8 votes
2 answers
2k views

What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
Devashish Sonowal's user avatar
0 votes
1 answer
414 views

Sufficient conditions for an asymptotic compactness

This question relates a theory of Mosco convergence. Let $X$ be a compact metric space, and $\mu$ a Borel measure on $X$. A symmetric bilinear form $(\mathcal{E},\text{Dom}(\mathcal{E}))$ on $L^2(X,\...
sharpe's user avatar
  • 721
3 votes
1 answer
266 views

If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings converge uniformly on compact sets?

Let $u_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi_\delta$ a family of standard mollifiers with compact support. Hörmander claims in The Analysis of Linear Partial ...
Carlos Esparza's user avatar
1 vote
0 answers
78 views

A question about extension problem related to fractional laplacian

I am studying the article "An Extension Problem Related to the Fractional Laplacian" by L. Caffarelli and L. Silvestre. link. At page 2, for a function $f\colon\mathbb{R}^n\to\mathbb{R}$, we ...
inoc's user avatar
  • 339
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 $...
cts12's user avatar
  • 51
4 votes
1 answer
1k views

Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
Giovanni Febbraro's user avatar
2 votes
0 answers
120 views

Taking limits in stochastic partial differential initial value problems

Background: A (stochastic) Cauchy problem I am interested in looks like this: $$ (1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
Mark's user avatar
  • 657
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 $(u_n)_n$ ...
Ayman Moussa's user avatar
  • 3,425
5 votes
1 answer
764 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 ...
Vrouvrou's user avatar
  • 277