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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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 ...
Ali's user avatar
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2 votes
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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 ...
Luis Yanka Annalisc's user avatar
3 votes
0 answers
66 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-\|...
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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{\...
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1 vote
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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 & = ...
Paul Joh's user avatar
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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 ...
Agustín Oyarce's user avatar
1 vote
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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}\...
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3 votes
1 answer
178 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{...
Bogdan's user avatar
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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 ...
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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\\ ...
Paul Joh's user avatar
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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 ...
Lorenzo Pompili's user avatar
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$T$ 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 $$ Tu=u\;\text{ on }\partial U $$...
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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 ...
Isaac's user avatar
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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|^...
Ilovemath's user avatar
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0 answers
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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 ...
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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) :=...
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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....
mathex's user avatar
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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$...
Ilovemath's user avatar
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1 answer
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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 ...
Ilovemath's user avatar
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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 ...
Keba's user avatar
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3 votes
2 answers
166 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 ...
leo monsaingeon's user avatar
2 votes
0 answers
168 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 ...
mnmn1993's user avatar
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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 \...
Isaac's user avatar
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1 vote
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88 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 ...
Lorenzo Pompili's user avatar
2 votes
0 answers
176 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^...
user113988's user avatar
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) ...
Guy Fsone's user avatar
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1 vote
0 answers
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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) ...
Nate River's user avatar
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5 votes
0 answers
154 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 ...
Juno Kim's user avatar
1 vote
0 answers
110 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 ...
Isaac's user avatar
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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 $C>0$ ...
mark's user avatar
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0 votes
1 answer
273 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 ...
Fei Cao's user avatar
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2 votes
0 answers
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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 \...
Keba's user avatar
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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 $D \in \mathbb{R}^{d \times d}$ ...
Fei Cao's user avatar
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4 votes
1 answer
331 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}^...
Keefer Rowan's user avatar
3 votes
1 answer
209 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)\...
Fei Cao's user avatar
  • 700
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 ...
Ilovemath's user avatar
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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 ...
Ilovemath's user avatar
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1 vote
0 answers
37 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 ...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
65 views

$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow

Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
Luis Yanka Annalisc's user avatar
1 vote
0 answers
84 views

$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition

Let $u$ be a solution of $$u' - \Delta u = 0 \quad\text{on $\Omega$}$$ $$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$ $$u(t=0)=u_0\quad\text{on $\Omega$}$$ where $\Omega$ is a bounded ...
BBB's user avatar
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3 votes
1 answer
131 views

The function $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m\,dy$ attains its maximum in $x=0.$

In this paper, Lemma 6, Pinsky proves that $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$ attains its maximum in $x=0.$ I didn't understand the proof very well, but ...
Ilovemath's user avatar
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2 votes
0 answers
110 views

Local smoothness of harmonic heat flow

Assume that $ (M,g) $ is a smooth closed manifold and $ \mathbb{S}^{L-1} $ is a unit sphere with dimension $ L-1 $ in $ \mathbb{R}^{L} $. Consider the equation of harmonic heat flow $$ \partial_tu-\...
Luis Yanka Annalisc's user avatar
2 votes
0 answers
62 views

Existence of Weak solution of a PDE with singular term

On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation $$ \begin{cases} u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
Ilovemath's user avatar
  • 585
4 votes
1 answer
428 views

Iterated Duhamel's formula for solutions of Boltzmann equation

My question comes from a computation in the paper Central limit theorem for Maxwellian molecules and truncation of Wild expansion. Specially, consider the following Boltzmann equation $$\frac{\partial ...
Fei Cao's user avatar
  • 700
1 vote
1 answer
108 views

Wild's sum for Boltzmann's equation

Consider the spatially homogenous Boltzmann equation $$\partial_t f_t = Q^+(f_t,f_t) - f_t.$$ A semi-explicit representation formula for solutions of this Boltzmann equation can be written as (see for ...
Fei Cao's user avatar
  • 700
1 vote
0 answers
98 views

Classical solution to logarithmic diffusion equation

Consider the one-dimensional logarithmic diffusion equation for $u: \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ : $$(\ast)\quad\quad \begin{cases} 2u_t = \big(\log(u)\big)_{xx} & \...
Fawen90's user avatar
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3 votes
0 answers
116 views

Wellposedness of this parabolic PDE

Consider a terminal-boundary value problem for $v: (t,x,y)\in [0,T]\times \mathbb R^2_+\to \mathbb R\ni v(t,x,y)$: $$ \begin{cases} v_t + \max(v_x,v_y)+ \frac 1 2 (v_{xx}+v_{yy})=0, & \forall (t,...
Fawen90's user avatar
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0 answers
42 views

This result of nonexistence of Hamilton–Jacobi equations in Lebesgue spaces is valid in domains?

In the article The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces Ben Artzi, Souplet, Weisller studies the equation \begin{equation} \left\{ \begin{array}{rll} ...
Ilovemath's user avatar
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4 votes
1 answer
93 views

A formula in harmonic heat flow

Assume that $ (M,g) $ and $ (N,h) $ are two smooth closed manifold and $ N $ is embedded isometrically into $ \mathbb{R}^K $ for some $ K\in\mathbb{Z}_+ $. Assume that $ u\in C^{\infty}(M\times\mathbb{...
Luis Yanka Annalisc's user avatar
0 votes
0 answers
27 views

How to deal with discontinous problem with numerical method?

I would like to consider how to deal with the indicator function in a PDE. For example, for the PME with 𝑚=5, the initial condition is the two-Box solution with the same height, namely $$ u_0(x)= 1 \...
Chet's user avatar
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