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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51 views

Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...
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52 views

Definiteness and infinite divisibility of kernels including heat semigroup

Let $P_{t}$ be the usual heat semigroup. Can one show (preferably) or disprove that for arbitrary $k \in \mathbb{R}_{>0}$ and $n \in \mathbb{N}$ we have \begin{equation} ...
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0answers
69 views

An Estimation on the Heat Kernel

I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get ...
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0answers
26 views

Find some conditons to $u_x$ is bounded

I am studying a inverse parabolic problem. $u_t-a(x, t, u, u_x)u_{xx}=b(x,t,u,u_x), (x,t)\in [0,\pi]\times[o,T]$ I want to find some conditions to $u$ and $u_x$ is bounded. I found in book: Second ...
4
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1answer
160 views

Reference for a Heat Process in a Wedge

I would like to ask about an explicit suggestion/reference for the following type of heat processes: Roughly, assume we have a "wedge" $W$ of the following form - a domain in $\mathbb{R}^n$ with a ...
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0answers
39 views

Weak solutions of linear parabolic PDEs and corresponding SDEs

It is well known that for an Stochastic differential equation (on the real line) of the form: $dX_t = \mu(X_t)dt + \sigma(X_t)dW$ where $W$ is the standard Wiener process, the transition probability ...
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0answers
55 views

Regularity of the heat equation: Neumann boundary conditions

I am looking for references to regularity estimates for the solution of a heat equation with homogeneous Neumann boundary conditions on $[0,T]\times D$ for some smooth domain $D\subseteq \mathbb{R}^3$ ...
5
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1answer
69 views

Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$. Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...
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335 views

Properties of the solution of the heat equation

Note 1: the following question has been post on Math Stackexchange here but receive no respond. So I post it here to get more attention. Note 2: This is my research problem, but the original problem ...
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0answers
33 views

Another proof of the comparison principle for PDEs by the theory of viscosity solutions

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions. So I am now at the end of my tether becasuse I can not ...
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69 views

A heat equation approach to the perturbation of vector field with center

Edit: According to the comment of Willie Wong I realize that the previous version was trivial. I thank him for his comment. Now I revise it. We consider the heat equation $$U_{t}=\Delta ...
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2answers
249 views

Real analyticity of solution of heat equation

Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
3
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1answer
154 views

Scaling properties of the Hölder estimate for heat equation

Lately, I have been interested in scaling properties of parabolic equations, and this question is related to an earlier one I asked about Harnack constants. Let $Q(R) := Q(R^2,R) = B(0, R) \times ...
2
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1answer
113 views

Heat equation: impact of the diffusion coefficient on the Harnack constant

Consider the heat equation $$ u_t - div[a(x,t) \nabla u] =0,\quad (x,t) \in B(r) \times [-r^2, 0] \subset \mathbb R^{d+1} $$ for a Hölder continuous coefficient $a(x,t)$ satisfying $$ 0<C_o \le ...
2
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0answers
58 views

Reference/proof for parabolic Holder spaces property

Suppose that a function $u:[0,1]\times[0,T]\to\mathbb{R}$ belongs to the parabolic Holder space $C^{2+\alpha,1+\alpha/2}$, for $\alpha\in(0,1)$. What can be said about $u_x=\partial_x u$? I am not ...
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0answers
33 views

Behavior of fundamental solution for parabolic equation on non compact complete riemannian manifold

Suppose that M is a complete noncompact Riemannian manifold. What is the necessary and sufficient condition that an operator on $L^{2}(M)$ comes from a smooth kernel that itself and all of the its ...
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0answers
54 views

Is there any theory of Hamilton-Jacobi system?

I am curious that is there any theory for (time-dependent) HJ system? I know for HJ equation, we have viscosity solution, which depends heavily on Maximal principle. However, for systems, this seems ...
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41 views

Is the heat kernel satisfies the heat equation in viscosity sense?

Let us see the heat kernel \begin{equation} k(x,t)= \begin{cases} (4\pi t)^{-\frac{n}{2}}\exp\{-\frac{1}{4}t^{-1}|x|^{2}\},t>0\\ 0,t\leq0. \end{cases} \end{equation} It is easy to see that $k\in ...
0
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3answers
158 views

Fundamental solution for a parabolic PDE with constant coefficents

[Cross posting http://math.stackexchange.com/questions/1374384/fundamental-solution-for-a-parabolic-pde-with-costant-coefficents ] I don't know if this question is more appropriate in Mathematics and ...
2
votes
2answers
132 views

References for non-zero boundary value problem

I studied linear elliptic, parabolic and hyperbolic PDEs (boundary/initial value problem) in terms of existence, uniqueness and regularity. I studied always, following Evans book "PDE", the case with ...
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1answer
81 views

The hypoellipticity of a heat-like operator

I am aware that the heat operator (on a smooth manifold) is hypoelliptic. I am also aware that there are manifolds on which the Schrödinger's operator (with a $\Bbb i = \sqrt {-1}$ multiplying $\frac ...
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2answers
67 views

Maximum principle for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega)$, we denote by $S(t)f$ the solution of the equation $$ ...
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0answers
70 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ ...
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2answers
223 views

Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...
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0answers
74 views

Existence of Solution steady navier stokes with do nothing outflow condition

We consider the stationary navier stokes equation with mixed boundary conditions $$ \begin{align} -\nu\Delta u +(u\cdot\nabla) u + \nabla p &= 0\ \textrm{in}\ \Omega \\ div\ u&=0\ \textrm{in}\ ...
2
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1answer
378 views

Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function. The classical set up is the following : If $(M,g)$ is a riemannian manifold then we could consider the ...
0
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0answers
60 views

Uniform bound in Faedo-Galerkin method with time-dependent weight in inner product

Let $v_j$ be an orthonormal basis for $V=H^1(\Omega) \subset L^2(\Omega)$ which is orthogonal in $L^2(\Omega)$. Let $w:[0,T]\times\Omega \to \mathbb{R}$ be a time-dependent weight which is smoooth ...
5
votes
2answers
246 views

Well-posedness of Fokker-Planck equation

Consider the following equation on $[0,T]\times\mathbb{R}^n$ \begin{eqnarray} &\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\ &\rho|_{t=0}=\rho^0, \end{eqnarray} where $V\in ...
3
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1answer
205 views

When does the cumulative distribution function solve the Kolmogorov backward equation?

For a diffusion $X$ define the cumulative distribution function for $X_T$ started with $X_t=x$: $$u(t,x):=E^{t,x}(1_{X_T\ge y})$$ Under what conditions does $u$ solve $X$'s Kolmogorov backward ...
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0answers
70 views

Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to $$ \frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$ for some elliptic differential operator $L$ where $f$ is not ...
3
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1answer
153 views

Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values. Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
5
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3answers
366 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
1
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1answer
100 views

Nodal sets under the heat flow

Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$ $u_t=\Delta u,$ where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of ...
1
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1answer
128 views

Strong maximum principle for the heat equation in non-cylindrical domains

let $u(t,x)$ be a bounded smooth solution of the heat equation $u_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: ...
6
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2answers
383 views

Alternative proof of Varadhan's formula on Riemann manifolds

Consider Varadhan's famous formula for the kernel of the heat equation on a manifold: $$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$ I do not have access to his 1967 two ...
7
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1answer
189 views

Mountain Pass theorem for minimization problems with constraints

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see ...
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0answers
70 views

solution to a parabolic PDE

I'm reading a paper where the following parabolic PDE is considered: $u_t(x,t)=u_{xx}(x,t)+b(x)u_x(x,t)+\lambda(x)u(x,t)$, with boundary conditions $u_x(0,t)=qu(0,t) \text{ and } u(1,t)=\int_0^1 ...
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0answers
71 views

Question on the local existence theory for the classical solution for the incompressible fluid dynamics equation

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} ...
3
votes
2answers
229 views

Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
2
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0answers
105 views

long time existence of a nonlinear parabolic equation

I am thinking about a geometric problem which boils down to the following parabolic equation: Suppose $u=u(r,t)$, $r$ is defined on $[0,1]$ and $t>0$ $$\begin{cases}\displaystyle \frac{\partial ...
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1answer
231 views

Heat kernel upper bound on compact Riemannian manifold

Let $M$ be a compact Riemannian manifold without a boundary. Let $p_t(x,y)$ be the heat kernel. I am looking for a reference for the result: there exists a constant $C$ such that $$|p_t(x,y)| \leq C$$ ...
2
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1answer
175 views

Local Biot-Savart law in $B(x_o,r) \subset \mathbb R^2$

Let $u: \mathbb R^2 \to \mathbb R^2$ and let $\omega = \text{curl } u$ be the 2D vorticity of $u$, where $u, \omega \in L^2(\mathbb R^2)$ and $\nabla \cdot u = 0$. The classical Biot-Savart law states ...
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0answers
145 views

Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as: \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0 \end{equation*} where ...
2
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0answers
106 views

Existence of solution to weak form of linear equation with boundary integral (parabolic PDE)

Let $W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}$. Let $\gamma$ and $\xi$ denote the trace map and its right inverse. Does there ...
3
votes
2answers
99 views

Estimates on a heat process with fixed boundary data and zero initial conditions

Consider the following heat process: For a given (say, smooth) domain $\Omega$ on a closed manifold $M$ we construct $p(t,x):\mathbb R_+ \times \bar\Omega \rightarrow [0,1]$, so that $$ \partial_t ...
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0answers
66 views

Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$

Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
2
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1answer
239 views

Extensions in parabolic Hölder spaces

Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$. As one could find in G.M. Troianello "Elliptic Differential Equations and ...
2
votes
2answers
295 views

Curvatures preserved under the Kahler-Ricci flow

Maybe it is a trivial question. Is there any obvious reason that non-negative holomorphic bisectional curvature is preserved by (normalized) Kahler-Ricci flow, but non-negative Ricci curvature is not ...
3
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0answers
101 views

Reference for existence results for 2D forced viscous Burgers equation

I am looking for results concering the following parabolic PDE $$u\cdot\nabla u + \Delta u = F(x),$$ where $$u\colon\Omega\to\mathbb{R}^2,$$ and $\Omega\subset\mathbb{R}^2$ is a 2D domain (bounded or ...
2
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0answers
83 views

Properties of solutions of Parabolic type equations

Assume $u\in C([0,1],L^{2})$ satisfies the following Schrodinger equation $$ \partial_t u=i(\Delta u+Vu), \text{in} ~\mathbb{R}^n\times[0,1],\\ u(0)=u_{0}. $$ with $V=V_1(x)+V_2(x,t)$, where $V_{1}$ ...