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.
493
questions
4
votes
1
answer
311
views
Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
- 1,330
4
votes
1
answer
149
views
$L^2$ norm for solutions of evolution equations driven by different elliptic operators
Let $u$ be a solution of the heat equation
$$u_t - \Delta u = 0, \qquad t >0, \ x \in \mathbb T^d$$
and $v$ be a solution of the bi-harmonic heat equation
$$v_t +\Delta^2 v = 0, \qquad t >0, \ x ...
- 293
1
vote
0
answers
56
views
Elliptic principal eigenfunction analysis for Langevin dynamics with a varying source term
Consider the Kolmogorov forward equation for a Langevin dynamic:
$$\DeclareMathOperator{\Div}{div}
\begin{cases}
\dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\
\\
\displaystyle\int_{\...
- 11
4
votes
0
answers
102
views
The logistic elliptic equation
Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
- 1,330
3
votes
2
answers
221
views
Change of variables for obtaining a unitary group
Consider the following NLS:
$$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$
where $F(u):=(u + \bar{u} + |u|^2)u.$
In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
- 159
2
votes
0
answers
122
views
Parabolic maximum principle for non-compact manifold with boundary
Let $M:=\mathbb{R}^n\setminus \mathbb{D}^n$, where $\mathbb{D}^n$ is the open unit ball in $\mathbb{R}^n$ and $u\colon M\times [0,\infty)\rightarrow \mathbb{R}$ is solution of the following PDE
\begin{...
- 632
3
votes
0
answers
121
views
Is the normalized Ricci flow real analytic in the time variable?
Let $(M^n,g)$ be a closed Riemannian manifold. In this paper, B. Kotschwar proved that the Ricci flow $g(t)$ with initial condition $g(0) = g$ is real analytic with respect to the time variable, for $...
- 2,085
-3
votes
1
answer
98
views
Asking for reference about a relation related to Fourier transform [closed]
Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$
Could ...
- 159
3
votes
2
answers
292
views
Hamiltonian, energy, and conservation laws of nonlinear PDEs
In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
- 159
1
vote
0
answers
44
views
Are derivatives of the solution to parabolic PDEs dominated by gaussian densities?
Consider the parabolic PDE for $p:\mathbb R_+^2\to\mathbb R_+$
$$\partial_tp =a(t,x)\partial^2_{xx}p+b(t,x)\partial_x b+c(t,x)p,\quad \forall t,x>0 \quad (\ast)$$
and $p(0,\cdot)=p_0$ and $p(\cdot,...
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3
votes
0
answers
118
views
Holmgren's theorem on the boundary
Consider $\Omega$ a bounded Lipschitz domain, with $\gamma \subseteq \partial \Omega$ a $C^2$ manifold. I am interested in proving the following.
Let $u: \Omega\times [0,T]\rightarrow \mathbb{R}$ be ...
- 225
-1
votes
1
answer
78
views
A question about the commutator $[J^s,u]\partial_x u$
I am studying the use of the commutator for finding the estimate of energy. During my looking through many papers I found that this paper contains a possible typo. Here is the archive version which ...
- 159
3
votes
0
answers
44
views
An equality satisfied by the solutions to Kolmogorov forward and backward PDEs
Let $b: \mathbb R_+\times\mathbb R\to \mathbb R$ and $\sigma: \mathbb R_+\times\mathbb R\to (0,\infty)$ be functions as nice as possible (e.g. bounded and of bounded partial derivatives, and $\inf_{(t,...
- 1,230
1
vote
0
answers
68
views
Existence of a classical solution to some linear parabolic PDE with Dirichlet condition
Consider the following parabolic PDE (Fokker-Planck equation) for $u: \mathbb R_+\times\mathbb R_+ \to\mathbb R$:
$$\partial_t u(t,x) = \frac{1}{2(1+q(t))}\partial^2_{xx}\big(a(t,x)u(t,x)\big)-\frac{1}...
- 1,230
1
vote
1
answer
391
views
Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space
I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$:
$$
\partial_t f = {div} \left [\left( \...
- 63
4
votes
0
answers
143
views
Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
- 1,230
0
votes
0
answers
58
views
Sign of $\partial_x(\sigma^2p)(t,0)$ with $p$ solving some Fokker-Planck equation
Let $\rho : \mathbb R_+\to\mathbb R_+$ be a density (as good as possible), i.e.
$$\int_0^\infty \rho(x)dx=1.$$
Consider the parabolic PDE
$$\partial_t p(t,x) = \frac{1}{2}\partial_{xx}^2\big(\sigma^2(...
- 1,230
2
votes
1
answer
186
views
Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?
Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?
I have seen many papers on Liouville's equation $\Delta u=K e^{ u}$ when $K>0$, such as enter link description ...
- 729
0
votes
1
answer
106
views
FEM based solution to parabolic problem
Consider the problem
$$
\begin{cases}
u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega
\end{cases}
$$
...
- 225
0
votes
1
answer
76
views
Changing the system of PDE by diffeomorphism and differentiate a composition
This problem comes from the book Hamilton's Ricci flow.
Given a smooth functional $f$, and following system. $$\partial_t f=-(\Delta f+R)$$ If there exist a 1 parameter family of diffeomorphism $\Psi(...
- 263
1
vote
0
answers
37
views
Parabolic theory for singular coefficients on bounded domains (Reference Request)
In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems.
Is ...
- 111
3
votes
0
answers
101
views
On the derivatives of the solutions of the heat equations with Neumann boundary condition
Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
- 701
0
votes
0
answers
47
views
How was this heat semigroup estimate made in a paper on reaction–diffusion systems?
In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage
$$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$
Here $S(t)$ denotes the heat ...
- 585
1
vote
0
answers
18
views
Stabilization of the second BVP solutions for nondivergence parabolic equations
Let $Q\subset \mathbb R^n$ be a bounded domain with smooth enough boundary $S$.
For a uniformly parabolic operator
$$
Lu=u_t-\sum_{i,j=1}^n a_{ij}(x)\partial_{ij}u-\sum_{i=1}^n b_{i}(x)\partial_{i}u
$...
- 2,645
0
votes
0
answers
83
views
Partial derivative of the Bessel's operator
Let $J^s = (I- \Delta)^{\frac{s}{2}}$ where $\Delta$ is the Laplacian, and $w(x,y) \in L^2(\mathbb{T}^2)$. During my study to the paper, https://arxiv.org/pdf/1809.02027.pdf, the author stated that
$$\...
- 159
2
votes
1
answer
175
views
Parabolic Schwarz lemma
Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
- 59
1
vote
0
answers
82
views
Is there any class of initial data for which the heat semigroup is increasing in time?
Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is
...
- 585
5
votes
0
answers
111
views
$L^p$ estimates for linear parabolic pdes
Let $u$ solve the linear parabolic equation
$$
u_t - \Delta u = f \text{ on } \Omega \times (0,T)
$$
with initial condition $u(0)=u_0$ and homogeneous Dirichlet boundary condition on $\partial \Omega ...
- 248
6
votes
1
answer
270
views
Short time existence for fully nonlinear parabolic equations
I am trying to assert short time existence for a fully nonlinear equation of the general form
\begin{equation}
\begin{cases}
u_t = F(x,u,Du,D^2u) & \text{in }(0,T)\times\Omega\\
u(\cdot,0) = u_0(\...
- 61
2
votes
0
answers
115
views
Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
- 121
0
votes
0
answers
142
views
Compact embedding of anisotropic Sobolev space
I am wondering if the embedding from $W^{2,1}_p(\Omega \times [0,T])$ to $C^{\alpha,\alpha/2}(\Omega \times [0,T])$ is compact, for some suitable domain, $p$ and $\alpha$. I have found some results. I ...
- 44
1
vote
0
answers
86
views
Reference for unique classical solution to quasilinear uniformly parabolic PDEs
In this post, the author mentioned that "we know there is a unique classical solution (see the references below, for example)". I have tried to read the two references the author provided, ...
- 11
4
votes
1
answer
289
views
Reference request: continuity of the derivatives of the (fundamental) solution to a parabolic equation
Consider the parabolic equation in $p: \mathbb R^2\to\mathbb R$
$$\partial_t p + b(t)\partial_x p + D(t,x)\partial^2_{xx}p=0,$$
where $b$, $D$ are nice enough functions. I look for the continuity of ...
- 1,230
1
vote
0
answers
103
views
Uniqueness of the solution to some parabolic PDE
Consider the system
$$
\begin{eqnarray}
\partial_t p(t,x) + b(t)\partial_x p(t,x) - \partial_{xx}^2\left(\frac{\sigma(t,x)^2p(t,x)}{2\big(1+\alpha(t)\big)^2}\right) &=& 0, & \forall t>0,...
- 1,230
1
vote
1
answer
102
views
Local boundedness for Cauchy problem
Consider the Cauchy problem
$$\left\{\hspace{5pt}\begin{aligned}
&-\dfrac{\partial u }{\partial t}
+a\dfrac{\partial^2 u}{\partial x^2}
+b \dfrac{\partial u }{\partial x}
+c u
= f(u) \leq 0& ...
- 44
4
votes
0
answers
304
views
Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
- 213
1
vote
0
answers
60
views
Examples of reaction-diffusion systems with analytical solutions
I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:
$$\...
- 1,330
1
vote
0
answers
198
views
Is this generalization of differentiable manifolds to mixed dimensions a known object?
Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls.
This would amount to studying the ...
- 131
1
vote
0
answers
67
views
Parabolic/Elliptic equation with nonlinear gradient term
Let $a\in (0,1)$ and $(0,1) \subset \mathbb{R}$, we consider the below equation in $(0,1) \times (0,T)$
$$ \partial_t u -\partial_x^2 u - \dfrac{1}{|u|^a} \partial_x u =f(x).$$
And $u(x,0)=x^{1/a}$ ...
- 44
0
votes
1
answer
57
views
Set invariance for differential inclusions applied to PDES?
This question is somewhat related to this one that I posted a while back on MSE, but the context has slightly changed since then. My question here relates to the consequences of a result in Weinberger'...
- 121
2
votes
0
answers
198
views
Dependency of fundamental solution on coefficients of heat equation
Let $b: \mathbb R_+\to\mathbb R_+$ and $\sigma: \mathbb R_+\times \mathbb R\to\mathbb R_+$ be Lipschitz and bounded. Assume further $\sigma$ is elliptic, i.e. $\inf_{(t,x)}\sigma(t,x)>0$. For each $...
- 1,230
1
vote
0
answers
43
views
Time evolution of Wigner transform
I am studying the Hartree equation for N-particles for the first time and things are not clear to me. Given the density matrix
$$\gamma_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) =
\begin{cases}
\int \...
- 159
4
votes
0
answers
102
views
$\mathcal{C}^1(\overline{\Omega})$ gradient bounds for the Dirichlet problem of the heat equation on general domains
I am studying the heat equation on a general bounded domain $\Omega \subset \mathbb{R}^+ \times \mathbb{R}^n$ with continuously differentiable Dirichlet data $\phi$ on the boundary,
$$
\left\{
\begin{...
- 173
0
votes
1
answer
137
views
Proof of vanishing viscosity error rate
Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$.
What is a ...
user139844
2
votes
0
answers
80
views
Solution verification of some PDE with an additional condition
Provided a probability density $\rho:\mathbb R_+\to\mathbb R_+$ (as nice as possible), consider the PDE
$$
\begin{cases}
\partial_t p = \dfrac{\partial_{xx}p}{2\big(1+m(t)\big)^2} - \partial_x p, &...
- 1,230
1
vote
0
answers
115
views
Regularity of solutions to heat equations
Let $d$ denote a positive integer. Let $f$ be a positive function on $\mathbb{R}^d$.
We also assume that $f$ is bounded above and below. That is, there exists $C>0$ such that $C^{-1}\le f(x)\le C$, ...
- 701
2
votes
1
answer
364
views
PDE interpretation of some properties of the solution to Fokker–Planck equations
Consider
$$X_t=X_0 + \int_0^t b(s)ds+ \int_0^t \sigma(s)dW_s,\quad \forall t\ge 0,$$
where $X_0\ge 0$ is a random variable of density $\rho$, $(W_t)_{t\ge 0}$ is an independent Brownian motion and $b,\...
- 1,230
4
votes
1
answer
452
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 $p(0,\cdot)=\rho$...
- 1,230
2
votes
1
answer
76
views
Parabolic system with coupling in the diffusion
Let's consider the parabolic system
$$
\begin{cases}
u_t - \Delta u -a\Delta(uv) = 0 \\
v_t - \Delta v - b\Delta(uv) = 0
\end{cases}
$$
with $a,b >0$. What is the name of this system? Are there ...
user139844
5
votes
1
answer
236
views
Parabolic equation with Cauchy boundary condition
Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. I would like to obtain the existence of the problem
\begin{equation}
\left\{\...
- 44