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
5
votes
2
answers
344
views
Functional decaying under the heat flow (?)
Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.
For any positive function $v$, I set
$$
J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g.
$$
...
- 1,253
3
votes
1
answer
619
views
Rate convergence of the heat equation as diffusion tends to zero
Is there a good reference for the following problem? Consider any smooth bounded domain $\Omega$ and solve the heat equation
\begin{align}
\partial_t u^\kappa &= \kappa \Delta u^\kappa,\\
u^\...
- 61
3
votes
2
answers
1k
views
Reference for De Giorgi-Nash-Moser theory
I am interested in Holder regularity for equations of the form
$$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.
This was proved in the seminal paper of John Nash ...
- 483
1
vote
2
answers
667
views
Schauder regularity heat equation
Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
- 291
1
vote
1
answer
113
views
Vorticity equation for generalized Naiver Stokes equations
In $3D$ or $2D$, I can get the vorticity equation for the incompressible NSE; however, what's the vorticity equation for the generalized NSE? Does the fractional laplacian commute with the curl?
- 11
1
vote
0
answers
84
views
Parabolic (heat) PDE Green's function spatial asymptote at infinity
Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its ...
- 2,169
3
votes
0
answers
316
views
Critical spaces and energy estimate in NS equation [closed]
There is a ‘rescaling transformation’ that is particularly significant for
the Navier–Stokes equations when they are posed on the whole space, but is
also important in the local regularity theory.
...
- 139
4
votes
0
answers
181
views
Relationships between fractional Sobolev space, Bessel spacse and Hajłasz–Sobolev space
It is known that for $\alpha\in(0,1)$ and $p>1$,
the fractional Sobolev space $W^{\alpha,p}(R^n)$ is defined by
$$
W^{\alpha,p}(R^n):=\{f\in L^p(R^n):\int_{R^n}\int_{R^n}\frac{|f(x)-f(y)|^p}{|x-y|^...
- 411
2
votes
0
answers
102
views
Inhomogeneous heat kernel estimates
I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation
$$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$
where $\...
- 9,591
0
votes
1
answer
268
views
Expected properties for a PDE whose solution is supposed to be something that doesn't exist
My understanding of Lecture #33, 34: The Characteristic Function for a Diffusion:
As an alternative to directly computing the characteristic function of a random variable $X_t$ in a stochastic ...
- 247
2
votes
0
answers
92
views
Reference request: $|\partial_t u - \Delta u|\in L^p(\Omega^T)$ implies Holder estimate
I am currently reading a paper regarding harmonic flow between Riemannian manifolds. Let $\Omega$ be a Riemannian domain of dimension $2$, $\Omega^T=\Omega\times[0,T]$. The equation is
$$
\partial_t ...
- 1,245
6
votes
0
answers
372
views
Linear PDE with non constant coefficients and properties of Green's Function
Lots of information is available about Poisson's PDE $\operatorname{div}(\operatorname{grad}(u(\vec{x}))))=f(\vec{x})$. However it is hard to find information about the more generalized case
\begin{...
- 61
3
votes
1
answer
309
views
Parabolic Regularity with Neumann B.C
Consider the parabolic problem in the cylinder of base $B$, the unit ball,
$$
\partial_t u -\text{div}\left( A(x) D u +F(t,x)\right)=0 \text{ in } (0,T)\times B,
$$
with $(ADu +F)\cdot \nu=0$ on $(0,T)...
- 2,464
0
votes
0
answers
74
views
Looser condition for regularity for Neumann problems
If $u(x) = g(|x|)$ is a rotationally symmetric function in $\mathbb{R}^{n+1}$ then
$$\Delta u = g''(|x|) + n |x|^{-1} g'(|x|).$$
Let's say we are studying rotationally symmetric solutions to ...
- 856
6
votes
1
answer
830
views
Injectivity of a Fredholm operator
While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.
Question : Suppose $K$ is ...
- 386
3
votes
1
answer
695
views
Looking for access to McKean's original paper?
I'm looking for the PDF version/scan of Henry P McKean Jr.'s paper on propagation of chaos. The reference is as follows -
Propagation of chaos for a class of non-linear parabolic equations., In ...
- 155
0
votes
0
answers
89
views
Movement of a random walk in the limit (a particle in diffusion)
I asked this question in Math Exchange and obtained no answer.
Let $X(t)$ be a stochastic process in time such that $X(0)=0$ and, at each increment of time $\Delta t$, it can move $h$ units in space ...
- 141
2
votes
1
answer
450
views
Motivation behind the parabolic metric
I've been reading some papers about parabolic evolution problems between manifolds. We want to study the behaviour of maps from the domain $(0,T)\times \Omega$, where $\Omega\subset \Bbb R^m$, to some ...
- 1,245
0
votes
1
answer
634
views
Green's functions/fundamental solution to a non-constant coefficients pde
We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
2
votes
0
answers
109
views
Differentiable dependence on the data for parabolic equations
Let $g_{\lambda}$ be a one parameter family of Riemannian metrics, which are complete and with bounded curvature, on the unit disk, depending smoothly on the parameter $\lambda$. Let $\Delta_{\lambda}$...
4
votes
0
answers
94
views
One-dimensional harmonic map flow with low regularity
My question is the following:
What is the minimum regularity for a continuous loop $\gamma: S^1 \rightarrow M$ in a Riemannian manifold $M$ to have short-time existence for the harmonic map flow in ...
- 9,591
1
vote
1
answer
190
views
Generator of spacetime Markov Process is Parabolic?
Suppose one considers some non-autonomous SDE thereby the Markov transition function is not homogeneous. In order to "recover" some homogeneity, one can consider the "spacetime" or "lifted" ...
- 11
1
vote
0
answers
248
views
Classical solutions for parabolic PDE's
I am studying Ricci flow theory and the Ricci flow is not a parabolic equation. But there is some variant of it, called DeTurck-Ricci equation, that happens to be a parabolic PDE. So to argue ...
- 1,774
1
vote
1
answer
144
views
Mild solution of 2D surface quasi-geostrophic (SQG) equation
I was reading one of Kato's papers on Navier-stokes equations. A mild solution can be denoted as $u= e^{t\Delta}u_0 + \int_{0}^{t} e^{(t-s)\Delta} \mathbb P\nabla \cdot(u \otimes u)ds$, where $\mathbb ...
- 11
1
vote
1
answer
236
views
Infinitesimal generator of a semigroup with parameter
When we talk about evolution equation the first idea that comes to the mind is the semigroups theory. This theory deals with Cauchy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au,$$
...
- 545
1
vote
0
answers
322
views
Existence and Uniqueness of Solutions to Quasilinear Parabolic PDEs
Consider the following general form of a quasilinear parabolic PDE
$$ u_t = a(x,u,u_x)u_{xx} + b(x,u,u_x) \ \ \textrm{ for }-1<x<1, \tag{1}$$
with inhomogeneous boundary condition $u_x(x,t) = g(...
- 155
2
votes
1
answer
131
views
Some questions on parabolic function spaces
I remember I read those problems some place, but I cannot find it. Does anyone have any idea where I can find it?
If $X$ is a Banach space, then $(L^1(a,b;X))^*\cong L^\infty(a,b;X^*)$?
$X, Y$ are ...
- 323
3
votes
0
answers
238
views
Maximum principle for parabolic PDE's
I am reading Cao's paper about his proof on Calabi-Yau theorem and I am having two little questions that I shall post separately.
Let $M$ be a complex compact manifold of dimension $n$ and consider ...
- 1,774
4
votes
1
answer
627
views
Metrics $g_1\leqslant g_2$ implies the Ricci flow $g_1(t)\leqslant g_2(t)$?
Let M be a complete,n dimensional Riemannian manifold without boundary. Suppose $g_1,g_2$ are two metrics on M and $g_1\leqslant g_2$. Suppose that there exists $T>0$ such that for $i=1,2$, the ...
2
votes
1
answer
252
views
Growth at infinity of a solution to a parabolic PDE
Let us consider the equation:
\begin{align*}
(\partial_t - \Delta - b(t,x) \partial_x) u(t,x)& = f(t,x) \\
u(0,x) & = u_0
\end{align*}
defined on the whole real line (so in one dimension - but ...
- 377
2
votes
1
answer
271
views
Reference for the parabolic Cauchy problem on $\mathbb{R}^N$ or $\mathbb{T}^N$
I am searching for a reference for the general (uniformly) parabolic Cauchy problem of second order, that is
\begin{align*}
\partial_t u - \sum_{1\leq i,j\leq N}\partial_{x_j}(a^{ij}\partial_{x_i}) +...
- 2,710
4
votes
1
answer
2k
views
Crandall & Rabinowitz Theorem, bifurcation curves
Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type:
$$
F(\lambda,u) = \lambda u - G(u) = 0,
$$
where $G \in C^1(X,X)...
4
votes
0
answers
238
views
Space-time Poincaré inequality for solution of parabolic equation
If $u : \mathbb R^n \to \mathbb R$ is a smooth enough function then on any Euclidean $n$-ball $B_R$ of radius $R$ we have the very well-known Poincaré inequality
$$ \int_{B_R} |u - \bar u|^2 \le C(R,...
4
votes
0
answers
96
views
Existence and uniqueness of an elliptic equation coupled with a parabolic equation (mean curvature flow)
Given a parabolic equation on a simply connected smooth domain $\Omega(t)$ with boundary $\Gamma(t)$
$$
\Delta u = 1 \quad on \quad \Omega(t)
\\
\nabla u \cdot n + u = g \quad on \quad \Gamma(t)
$$
(...
- 41
6
votes
1
answer
660
views
Heat Equation with an integral boundary condition
I have been struggling with following Heat equation IBVP,
\begin{equation}
\frac{\partial v\left(x, t\right)}{\partial t} = \alpha \frac{\partial^2 v\left(x, t\right)}{\partial x^2}, \quad t \in \left(...
- 281
2
votes
0
answers
227
views
Existence and uniqueness of heat equation on domain with discontinuous boundary
as the title says i'm looking for references dealing with heat equation with discontinuous boundary conditions. More precisely, let $\lambda_{L},\lambda_{U}$ functions defined on the interval $[0,T]$ ...
- 91
2
votes
0
answers
88
views
Boundary regularity of solutions to semilinear heat equation
Consider the Cauchy IVP problem
$$u_t - \Delta u + f(t,x,u) = 0, \quad (t,x) \in (0,T) \times \mathbb{R}^n,$$
with initial condition $u(0,x) = g(x), \ x \in \mathbb{R}^n.$
Can you point out a ...
- 293
1
vote
0
answers
92
views
How can I show the principal symbol is not elliptic?
Let us assume the principal symbol of a nonlinear differential operator $E$ at a point $p$ is
$$\sigma_{E}(p):\Gamma (T^*M)\times \Gamma (T^*M)\to \Gamma (T^*M)$$
which acts as follows:
...
- 317
3
votes
1
answer
512
views
De Turck trick on mean curvature flow
I am reading the book Lecture on mean curvature flow by Xi-Ping Zhu.
Suppose $M^n$ is an n-dimension smooth manifold and $X(x,t):M^n \rightarrow R^{n+1}$ be a one-parameter family of smooth immersion....
- 44
1
vote
2
answers
481
views
Derivation of the volume preserving mean curvature flow
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Picture above is from
Huisken, Gerhard, The volume preserving ...
- 165
3
votes
0
answers
143
views
Prove the positivity of the subelliptic operator heat kernel
Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition at every point $x\in\mathbb{R}^{n}$, Let $W$ be a ...
- 265
1
vote
0
answers
106
views
Smoothness of the solution of 1D diffusion equation
How do I show that the solution of 1D diffusion equation is smooth for all t>0? I do know that in order to show a nonlinear PDE, for example Burger's equation, develops corners (instead of smooth ...
- 65
13
votes
0
answers
365
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. ...
- 856
3
votes
0
answers
96
views
Uniqueness of 1-dimensional heat-equation with dynamic boundary condition
My question regards uniqueness of the pair $(u(t, x), v(t))$ which
satisfy the following one dimensional time dependent heat equation
with a(n) (also time varying) Robin boundary condition at the ...
- 143
4
votes
2
answers
761
views
Heat equation close to the steady state
Let $u(t, x)$ be the unique solution of the heat equation on the unit interval with Dirichlet boundary conditions and initial data $u_0$:
$$
\left\{
\begin{array}{l}
\partial_t u(t, x) = \partial_x^2 ...
user105541
1
vote
0
answers
75
views
For what potentials is the heat operator with a potential term hypoelliptic?
If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat ...
- 5,282
1
vote
0
answers
154
views
Lyapunov stability for nonlinear PDEs
Where can I find a theorem about Lyapunov stability for the equation in Hilbert space?
$$
y' = Fy,
$$
where $F : V \to V'$ is a nonlinear operator , $y' \in L^2(0,T,V')$, $V$ is a Hilbert space.
...
- 243
1
vote
1
answer
171
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum II
This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions.
$$
\partial_t P = L^* P \tag1
$$
$$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^...
- 2,169
2
votes
1
answer
304
views
Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum
How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
- 2,169
5
votes
2
answers
903
views
Monograph on harmonic analysis with applications to PDEs
"There are two ways to teach mathematics, namely the systematic way and the application-oriented way"- E. Zeidler
I'm a fresh researcher on PDEs, especially interested in evolution equations in ...
- 407