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.
488
questions
4
votes
2
answers
331
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
84
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
$$
\partial_tu-\Delta ...
3
votes
0
answers
66
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\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-\|...
2
votes
0
answers
68
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
72
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
52
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 ...
1
vote
0
answers
55
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:
$$\begin{cases}\...
3
votes
1
answer
178
views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem:
$$\begin{...
2
votes
0
answers
148
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 ...
4
votes
1
answer
169
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
105
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 ...
1
vote
0
answers
70
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
$$
Tu=u\;\text{ on }\partial U
$$...
1
vote
1
answer
105
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
$$C\|\phi\|_X (e^{-|x|^...
0
votes
0
answers
39
views
Characterising optimal majorising Lyapunov function for Markov semigroup
Fix a space $\mathcal{X}$, a Markov process on that space with infinitesimal generator $L$, and a positive function $g : \mathcal{X} \to \mathbf{R}_+$. I don't want to assume too much more about the ...
4
votes
0
answers
120
views
Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$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 ...
0
votes
0
answers
14
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 ...
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 ...
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 ...
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
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 ...
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^...
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) ...
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
$$T_\varepsilon f(x) := \frac{1}{|B_\varepsilon (x)|} \int_{B_\varepsilon (x)} f(y) ...
5
votes
0
answers
154
views
Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)
Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$
$$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$
representing the Fokker-Planck evolution equation for the ...
1
vote
0
answers
110
views
Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?
I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation.
However, detailed proofs can ...
6
votes
0
answers
147
views
Gaussian lower heat kernel bounds on non-convex bounded domain
I am looking for a proof the following theorem.
Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
0
votes
1
answer
273
views
Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)
Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
2
votes
0
answers
58
views
Are solutions of the forced Navier–Stokes equation less regular than those of the Stokes equation?
Let $\Omega \subset \mathbb R^3$ be a smooth, bounded domain and $T > 0$. Let us consider
\begin{align*}
\begin{cases}
u_t + \kappa (u \cdot \nabla) u = \Delta u + \nabla P + f(x, t), \quad \...
2
votes
0
answers
73
views
Verify the explicit solution formula for a degenerate Fokker-Planck equation $\partial_t u = \nabla\cdot(D\,\nabla u + Cxu)$
Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$
$$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$
where $D \in \mathbb{R}^{d \times d}$ ...
4
votes
1
answer
331
views
Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight
The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form
$$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
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) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
1
vote
0
answers
37
views
regularity theory of parabolic equations in Heisenberg group
I'm trying to understand if there are regularity results for mild solutions of partial differential equations in Heisenberg group. In this paper (Theorem 1.3 (iii) and proof of Theorem 1.1) the author ...
0
votes
0
answers
39
views
Are there results of parabolic regularity in Heisenberg groups?
The $(2N +1)-$dimensional Heisenberg group $\mathbb{H}^N$ is the space $\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the ...
1
vote
0
answers
37
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
2
votes
0
answers
65
views
$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
1
vote
0
answers
84
views
$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$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 $\Omega$}$$
where $\Omega$ is a bounded ...
3
votes
1
answer
131
views
The function $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m\,dy$ attains its maximum in $x=0.$
In this paper, Lemma 6, Pinsky proves that $H(x) =(4\pi t)^{-n/2} \int_{\mathbb{R}^d} e^{\frac{-|x-y|^2}{4t}}(1+|y|)^m \, dy$ attains its maximum in $x=0.$ I didn't understand the proof very well, but ...
2
votes
0
answers
110
views
Local smoothness of harmonic heat flow
Assume that $ (M,g) $ is a smooth closed manifold and $ \mathbb{S}^{L-1} $ is a unit sphere with dimension $ L-1 $ in $ \mathbb{R}^{L} $. Consider the equation of harmonic heat flow
$$
\partial_tu-\...
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
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
4
votes
1
answer
428
views
Iterated Duhamel's formula for solutions of Boltzmann equation
My question comes from a computation in the paper Central limit theorem for Maxwellian molecules and truncation of Wild expansion. Specially, consider the following Boltzmann equation
$$\frac{\partial ...
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 ...
1
vote
0
answers
98
views
Classical solution to logarithmic diffusion equation
Consider the one-dimensional logarithmic diffusion equation for $u: \mathbb R_+ \times [0,1]\ni (t,x)\to u(t,x)\in \mathbb R$ :
$$(\ast)\quad\quad
\begin{cases}
2u_t = \big(\log(u)\big)_{xx} & \...
3
votes
0
answers
116
views
Wellposedness of this parabolic PDE
Consider a terminal-boundary value problem for $v: (t,x,y)\in [0,T]\times \mathbb R^2_+\to \mathbb R\ni v(t,x,y)$:
$$
\begin{cases}
v_t + \max(v_x,v_y)+ \frac 1 2 (v_{xx}+v_{yy})=0, & \forall (t,...
0
votes
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}
...
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{...
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 \...