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.
491
questions
5
votes
1
answer
145
views
Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
11
votes
1
answer
824
views
Modified energy method for transformed Fokker-Planck equation (tricky integration by parts…)
I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed ...
1
vote
1
answer
96
views
Reference to log-transition-density of a diffusion process
Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$
with $b$, $\sigma$ smooth, $\xi$ absolutely ...
2
votes
0
answers
60
views
Uniqueness of solution to Cauchy problem with quadratic nonlinearity
Consider the non-linear differential operator
$$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$
For $U\subset\...
0
votes
0
answers
138
views
When linear strongly elliptic operators are invertible?
I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like:
A linear differential operator, $$A : W^{...
1
vote
1
answer
140
views
first order derivative of the parabolic equation
Assume $b, \ell \in C_b^{1,2}(\mathbb R^2)$. We consider parabolic PDE
$$(P1)\quad \partial_t v = b \partial_x v + \partial_{xx} v + \ell, \ \forall (t, x) \in \mathbb R^+\times \mathbb R; \quad v(0, ...
4
votes
0
answers
110
views
Is there a better reference for existence/regularity for parabolic PDEs (and systems) than the book of Ladyzenskaja, Solonnikov, Uralceva?
The book of Ladyzenskaja, Solonnikov, Uralceva contains almost everything most people need yet the typesetting and notation is disgusting to the eye. Is there any better text that covers the same type ...
2
votes
1
answer
102
views
Explicit solution of a free boundary problem for heat equation
Consider the free boundary problem
$$
\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\
u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\
u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...
1
vote
0
answers
41
views
classical solution of nondegenerate HJB equation
Let $b\in C(\mathbb R)$ and $L \in C_b^2(\mathbb R)$. Consider an equation
$$v_t (x, t) + \inf_{a\in A} \{b(a) v_x(x, t) + a^2 \} + v_{xx}(x, t) + L(x) = 0, \hbox{ on } \mathbb R \times (0, 1)$$
with ...
4
votes
0
answers
183
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 ...
1
vote
0
answers
82
views
Solution existence for two-dimensional parabolic PDEs
I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system
$$
f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
5
votes
1
answer
437
views
Backward uniqueness for a heat equation with a drift
Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...
2
votes
0
answers
144
views
Integral estimate for the solution of the heat equation
Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality?
$$
\int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
1
vote
0
answers
135
views
Regularity theory for parabolic PDEs in fractional Sobolev spaces
I am trying studying on my own parabolic PDEs throughout the book "Linear and Quasi-linear Equations of Parabolic Type" by Vsevolod A. Solonnikov, Nina Uraltseva, Olga Ladyzhenskaya and the ...
0
votes
1
answer
235
views
Scaling argument for the heat equation in a bounded domain [closed]
We want to study the long time behavior of the heat equation $u_t - u_{xx} = 0$ in the domain $[-1,1]$. Now consider the rescaling $u^{\epsilon} = u(x/\epsilon, t/\epsilon^2)$. Then
$u^\epsilon$ ...
2
votes
0
answers
55
views
Stability of weak solutions to quasilinear parabolic equations
Consider the quasilinear operator $A(x,t,\nabla u)$ satisfying $$A(x,t,\nabla u).\nabla u \geq C_0 |\nabla u|^p$$ and $$|A(x,t,\nabla u)| \leq C_1 |\nabla u|^{p-1}$$ where $1<p<\infty$. Note ...
8
votes
1
answer
252
views
Positive solutions for semilinear parabolic equations
Let $X$ be a Banach lattice. Consider the system
$$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$
where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $...
1
vote
0
answers
142
views
Rigorous error estimate for semi-discrete heat equation in bounded domain
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of
$$
\begin{cases}
\partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\
u_h=0 &\text{ in } \...
3
votes
2
answers
876
views
Gradient $L^p$ estimates for heat equation
I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...
1
vote
0
answers
74
views
Very weak solution to parabolic PDE (pointwise a.e. in time with time derivative on test function)
Consider the parabolic PDE
$$u' + Au = 0$$
as an equality in $L^2(0,T;V^*)$ for some Hilbert space $V$ with $A\colon L^2(0,T;V) \to L^2(0,T;V^*)$ a coercive, bounded linear operator. Here $u'$ is the ...
2
votes
1
answer
357
views
Regularity on the boundary for the heat equation with linear source
This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...
2
votes
1
answer
215
views
Continuity of solution of a parabolic PDE w.r.t. system parameters
If we have a system of PDE of the form:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (...
2
votes
1
answer
449
views
Some properties of fractional Dirichlet heat kernel
Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition:
\begin{equation}
\...
4
votes
2
answers
357
views
Angle of analyticity of semigroup
Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ?
For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. ...
2
votes
1
answer
420
views
Existence of weak solutions of a parabolic PDE
Assume that $\Omega\subset\mathbb{R}^n(n\geq3)$ is a bounded open set with smooth boundary, $\varphi\in H^1_0(\Omega)$, $F(t)$ is differentiable in $\mathbb{R}$ and $F'$ is bounded. Given a PDE
$$
\...
2
votes
1
answer
284
views
An inequality for abstract Cauchy problem
Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...
1
vote
0
answers
99
views
Confusion optimal control abuse notation
I'm currently reading this paper describing a numerical scheme for the approximating optimal policy of a stochastic control problem. However, I run into a confusion directly on the first page where ...
3
votes
1
answer
228
views
Inequality for initial data
I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...
3
votes
1
answer
659
views
Reference request: Schauder estimate in the space variable for parabolic equations
Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$.
Let $$u\...
1
vote
1
answer
197
views
Analytical solution to inhomogeneous parabolic PDE [closed]
I would be thankful to anyone who can present an analytical solution to the following inhomogeneous PDE equation:
$$\frac{\partial{u}}{\partial{t}}= \alpha\frac{\partial^2{u}}{\partial{x^2}}-ku$$
$$...
2
votes
1
answer
386
views
Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations
I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801.
In Lemma 2.1, the authors said for any $\phi \in L^{\...
4
votes
0
answers
107
views
Biharmonic heat flow on compact manifolds
Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...
5
votes
1
answer
466
views
Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients
Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...
6
votes
0
answers
174
views
Reference request: Existence and regularity for parabolic PDEs with smooth coefficients on compact manifolds with boundary
I'm looking for a reference for a statement like:
Let $M$ be a $n$-dimensional smooth compact manifold with smooth boundary $\partial M$. In coordinates, let $\mathcal L$ have the form
$\mathcal L ...
1
vote
1
answer
86
views
A symmetric parabolic second order PDE
Here I want to solve a second order PDE symmetrically depending on two variables
$$
3(\partial_{\alpha\alpha}f+\partial_{\beta\beta}f-2\partial_{\alpha\beta}f)+2(\cot\alpha)\partial_\alpha f +2(\cot\...
5
votes
3
answers
514
views
Reference request: Long-term behaviour of the heat equation for bounded initial data
Let us consider the heat equation
\begin{align*}
\frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\
u(0,x) & = f(x)
\end{align*}
on the whole space $\mathbb{R^d}$. If $f \in L^p := L^...
2
votes
0
answers
33
views
Discrete maximum priniciple for parabolic operators
While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ...
4
votes
0
answers
602
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
4
votes
1
answer
212
views
Interesting questions for inverse parabolic problems
I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
1
vote
1
answer
121
views
Uniform Hopf Inequality
There is a Uniform Hopf Inequality as follow:
Let $\Omega \subset \mathbb{R}^n$, $n \geq 1$ denote a smoothly bounded domain. Also let $\rho(x)=\mathrm{dist}(x,\partial \Omega)$, the distance ...
4
votes
3
answers
608
views
Question about the regularity of fractional Heat equation
Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation:
$$
\...
3
votes
0
answers
41
views
Reference request: existence of strong solutions to a linear parabolic problem with mixed boundary conditions
on a domain $\Omega \subset \mathbb{R}^d$ with smooth boundary $\partial\Omega$ subdivided into two parts $\Gamma_D$ and $\Gamma_N$ I am considering the parabolic problem
$$
\partial_t u = \Delta u + ...
2
votes
0
answers
102
views
Energy functional continuous with respect of time $t$
I am studying a paper of Liu Yacheng which named "On potential wells and applications to semilinear hyperbolic equations and parabolic equations" it considers a nonlinear parabolic equation
\begin{...
2
votes
0
answers
59
views
Existence and uniqueness for semilinear problem
Consider the following problem:
$$-\Delta u + [(u)^+]^\alpha = 0,$$
where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...
1
vote
1
answer
246
views
PDE's : diffusion equation : polynomial diffusion coefficient
I'd like to find analytical solutions of that kind of differential equations :
$$\partial_t c = \partial_x (D(c)\partial_x c) $$
with $D(c)$ a polynomial. The trivial cas $D(c)=a$ with $a$ a ...
5
votes
1
answer
330
views
Exponential decay of solution in $L^p$ with $p>2$
Consider the following evolution equation
$$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 ...
1
vote
2
answers
112
views
Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}\le \hat C t^{1/q}$
Let $M$ be a $C^3-$compact manifold and $v \in W^{2,1}_p(M\times
[0,T])$ ($1\le p<\infty$) be the solution of:
$\begin{cases} \partial_t v-\Delta_{M} v=f(v), \quad M\times [0,T]\\
v(x,0)=v_0,...
2
votes
1
answer
593
views
Continuous embedding between parabolic Sobolev spaces
I was wondering whether the following continuous embedding theorem for parabolic Sobolev space is correct?
Let $I=[0,T]$ and $\Omega$ be a sufficiently smooth domain in $\mathbb{R}^n$, we consider ...
1
vote
0
answers
90
views
Name for a Particular (Parabolic) PDE
This is a cross post from MSE. The original question can be found here:https://math.stackexchange.com/questions/3248114/name-for-a-particular-parabolic-pde
Consider the following initial value ...
2
votes
0
answers
58
views
Time derivative in parabolic Hölder spaces
Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$.
Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-...