Questions tagged [parabolic-pde]

Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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1 vote
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39 views

Continuity of the constant in maximal Sobolev regularity

Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
1 vote
0 answers
134 views
+50

Specific type of PDE

While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices): $$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \...
-1 votes
0 answers
27 views

Numerical solution 2-D Poisson equation [closed]

I have some kind of Poisson equation: $-au_{xx}-bu_{yy} = f(x,y)$, $\ a,b>0$ and trying to solve it using FDM and Iteration methods (no matter which one, each method has the same problem). $$-a\...
2 votes
1 answer
214 views

Periodic solution for linear parabolic equation - existence, regularity

I am interested in proving the existence and regularity of solution to the following problem: $$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
4 votes
1 answer
447 views

A variant to the Fokker–Planck equation

Consider the PDE of $p(t,x)\ge 0$ given as $$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$ with initial and boundary conditions $p(0,\cdot)=\rho$...
4 votes
2 answers
336 views

Nontrivial invariant transformations for heat equations

It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by $$ v(r,\theta) = u(\frac{1}{r},\theta)$$ is also harmonic for $r>0$. Note that the Kelvin ...
2 votes
0 answers
88 views

Harmonic heat flow, formal and rigorous

Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of $$ \partial_tu-\Delta ...
3 votes
0 answers
69 views

Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
2 votes
1 answer
112 views

Reference: Result of interior parabolic regularity theory for Hamilton–Jacobi equations

Does anyone know the parabolic regularity result that Ben-Artzi used in the article The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces used to prove that the solution to the ...
2 votes
0 answers
69 views

Any solution of an evolution problem tends to a steady state in $L^2$?

I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
1 vote
1 answer
73 views

Second order differentiability of solution operator to nonlinear boundary value problem

I am currently reading the paper [1]. The paper deals with the BVP: \begin{align*} \partial_t u - \Delta u + u(u-a) & = -qu \text{ in } ]0,T\mathclose[ \times \Omega \\[6pt] \partial_{n}u & = ...
2 votes
0 answers
54 views

On improving the regularity of solutions to nonlinear parabolic pde

There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
0 votes
1 answer
278 views

Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)

Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
1 vote
0 answers
56 views

Parabolic regularity for weak solution with $L^2$ data

I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions: $$\begin{cases}\...
3 votes
1 answer
183 views

$L^{\infty}$ estimate for heat equation with $L^2$ initial data

Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem: $$\begin{...
4 votes
1 answer
354 views

Contractivity of Neumann Laplacian

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian. In ...
2 votes
0 answers
152 views

Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?

Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
3 votes
1 answer
199 views

Parabolic Sobolev inequality in Sobolev mixed norm spaces

Assume $p,q\in (1,\infty)$, $r\in [p,\infty)$, $s\in [q,\infty)$ and $$ 1<\frac{d}{p}+\frac{2}{q}=1+\frac{d}{r}+\frac{2}{s}. $$ Let $u\in C_c^\infty((0,1)\times B_1)$, where $B_1=\{x\in \mathbb{R}^...
4 votes
1 answer
171 views

Reference request: Solution to second order parabolic linear BVP belongs to $\mathcal{C}(0,T;H^1(\Omega))$

I am currently reading the paper [1]. In Theorem 3.1. b) the following boundary-value problem is given: \begin{align*} \partial_{t} y - \Delta y + g\cdot y = f \text{ in } ]0,T[ \times \Omega\\ ...
1 vote
0 answers
106 views

Global existence for large data in $H^{-1/2}(\mathbb R)$ of viscous Burgers' equation with external forcing

First, a quick summary of what to know about viscous (or dissipative) Burgers' equation $$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$ Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev ...
2 votes
1 answer
174 views

Parabolic Schwarz lemma

Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
1 vote
0 answers
71 views

$T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$

The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator $T: W^{1,p}(U) \rightarrow L^p(\partial U)$ such that $$ Tu=u\;\text{ on }\partial U $$...
6 votes
1 answer
996 views

Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
1 vote
1 answer
106 views

Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data

I already asked the question on MSE, and have tried to figure it out myself. But the problem seems trickier than expected, so I guess MO is a better place to ask.. For the sake of completeness, I ...
3 votes
2 answers
265 views

Is the passage in argument of existence solution of PDE correct?

The passage below is from a fixed point argument in Wang's paper enter link description here pg 566. At the end of the $I_1$ calculation, it somehow makes the following estimate $$C\|\phi\|_X (e^{-|x|^...
4 votes
1 answer
260 views

Elliptic regularity when the Lagrangian is possibly infinite

I want to solve variational problems of the form $$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$ where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
2 votes
0 answers
169 views

Regularity of linear Bellman equation

Let $f(x,t):B_1 \times [0,1] \to \mathbb{R}$ be Lipschitz function on both $x$ and $t$, $\varphi$ be Lipschitz function on both $x$ and $t$ on the parabolic boundary of $B_1 \times [0,1]$. Let $A$ be ...
13 votes
0 answers
362 views

Pseudolocality outside of geometric PDE?

In Ricci flow, the pseudolocality theorem says roughly that regularity in some region implies that as time goes on, there is some regularity in a smaller region. The first version is due to Perelman. ...
0 votes
0 answers
39 views

Characterising optimal majorising Lyapunov function for Markov semigroup

Fix a space $\mathcal{X}$, a Markov process on that space with infinitesimal generator $L$, and a positive function $g : \mathcal{X} \to \mathbf{R}_+$. I don't want to assume too much more about the ...
4 votes
0 answers
120 views

Trace-class heat semigroups

Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator. Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$ $$T_{\varphi}(u) :=...
1 vote
0 answers
77 views

Heat kernel and estimates

In the article by Hairer-Labbe (A simple construction of the continuum parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....
2 votes
0 answers
54 views

Can the regularity argument for the solution of a parabolic PDE in Pinsky's paper be generalized?

In this paper Pinsky shows existence, uniqueness and regularity for the problem $$ u_t=\Delta u-a(x) u^p |\nabla u|^q $$ where $a\in C^2( \mathbb{R}^d)$ satisfies the condition $ a(x)|\leq (1+|x|^2)^N$...
0 votes
1 answer
101 views

Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$

We consider the heat kernel $$ g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ). $$ The inequality (2.3) in this ...
3 votes
2 answers
167 views

Heat equation with nonlocal boundary condition

$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with ...
1 vote
0 answers
15 views

Parabolic equations and nonconvex domains

Is there a (system of) parabolic equation(s) where qualitative properties depend on whether or not the (say, smooth and bounded) domain $\Omega$ is convex? I am aware of a few cases where a proof ...
0 votes
0 answers
24 views

Detailed estimate of the magnitude for the constant appearing in maximal regularity of the inhomogeneous heat equation

For any $T \in (0,\infty)$ and the $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ together with $1< p,q < \infty$, let us consider the following Cauchy problem: \begin{equation} \partial_t U - \alpha \...
1 vote
0 answers
89 views

Burgers' equation with viscosity: modulational analysis and energy estimates for large data

I think my question applies to many PDE that admit stationary solutions or travelling waves. I will simply describe the setting that I am more familiar with, but the technique is standard and applies ...
2 votes
0 answers
182 views

Conditions for an existence of smooth solution to a parabolic PDE

I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset): \begin{equation*} u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
2 votes
0 answers
143 views

Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$

Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$ I would like to prove (or disprove) ...
5 votes
0 answers
157 views

Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)

Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$ $$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$ representing the Fokker-Planck evolution equation for the ...
1 vote
0 answers
94 views

Scaling limit of a discrete analogue of the heat equation

For $f \in L^1 (\mathbb R^d)$, given $\varepsilon > 0$, define the function $T_\varepsilon f$ on $\mathbb R^d$ by $$T_\varepsilon f(x) := \frac{1}{|B_\varepsilon (x)|} \int_{B_\varepsilon (x)} f(y) ...
1 vote
0 answers
114 views

Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?

I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation. However, detailed proofs can ...
6 votes
0 answers
147 views

Gaussian lower heat kernel bounds on non-convex bounded domain

I am looking for a proof the following theorem. Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
2 votes
0 answers
73 views

Verify the explicit solution formula for a degenerate Fokker-Planck equation $\partial_t u = \nabla\cdot(D\,\nabla u + Cxu)$

Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$ $$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$ where $D \in \mathbb{R}^{d \times d}$ ...
3 votes
0 answers
62 views

Are solutions of the forced Navier–Stokes equation less regular than those of the Stokes equation?

Let $\Omega \subset \mathbb R^3$ be a smooth, bounded domain and $T > 0$. Let us consider \begin{align*} \begin{cases} u_t + \kappa (u \cdot \nabla) u = \Delta u + \nabla P + f(x, t), \quad \...
4 votes
1 answer
335 views

Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight

The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form $$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
3 votes
1 answer
211 views

Reference request: analysis of a nonlinear Fokker-Planck type equation

It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
1 vote
0 answers
37 views

regularity theory of parabolic equations in Heisenberg group

I'm trying to understand if there are regularity results for mild solutions of partial differential equations in Heisenberg group. In this paper (Theorem 1.3 (iii) and proof of Theorem 1.1) the author ...
0 votes
0 answers
39 views

Are there results of parabolic regularity in Heisenberg groups?

The $(2N +1)-$dimensional Heisenberg group $\mathbb{H}^N$ is the space $\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the ...
1 vote
0 answers
38 views

Mixed boundary condition of parabolic equations

Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that $$ \partial\Omega=\partial\Omega_D\cup\partial\Omega_N, $$ where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...

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