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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3
votes
1answer
96 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
votes
1answer
89 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
votes
0answers
47 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 ...
1
vote
0answers
26 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 ...
1
vote
0answers
43 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 ...
0
votes
0answers
33 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
votes
3answers
137 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 ...
1
vote
2answers
71 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 ...
0
votes
1answer
65 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 ...
0
votes
2answers
60 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 $$ ...
1
vote
0answers
62 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: $$ ...
6
votes
1answer
136 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 ...
0
votes
0answers
54 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
votes
1answer
168 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
votes
0answers
49 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
193 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
votes
1answer
185 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 ...
1
vote
0answers
68 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
votes
1answer
139 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
votes
3answers
363 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
vote
1answer
93 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
vote
1answer
102 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: ...
5
votes
2answers
343 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
votes
1answer
160 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 ...
0
votes
0answers
42 views

Least square problems with binary variables

I want to solve the heat equation $T_t(x,t) = - L_x . T(x,t) + F(x,t)$ in an edge-weighted graph where $L_x = \sum_i x_i e_{ij}$ is weighted Laplacian matrix of the graph. Then I conclude to the ...
1
vote
0answers
68 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 ...
1
vote
0answers
64 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
228 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
votes
0answers
94 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 ...
1
vote
1answer
208 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
votes
1answer
148 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 ...
0
votes
0answers
133 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
votes
0answers
92 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 ...
2
votes
2answers
94 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 ...
0
votes
0answers
51 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
votes
1answer
235 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
272 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
votes
0answers
93 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
votes
0answers
71 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 $V_{1}$ ...
-1
votes
2answers
166 views

Motivation for weak solution of a PDE (initial condition)

The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations" When looking at a (nonlinear degenerate) PDE like $$ ...
1
vote
0answers
82 views

Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$. Consider the system, with $u^\epsilon, v^\epsilon ...
1
vote
0answers
62 views

References Request : Existence and Uniqueness for PDE which is “ALMOST (?)” Parabolic

I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case: \begin{equation} Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + ...
0
votes
1answer
124 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
6
votes
1answer
211 views

Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...
0
votes
1answer
139 views

Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question. I need some regularity results for the single and double layer heat potentials. If $\Gamma(t,x)$ is the fundamental ...
3
votes
0answers
168 views

Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
2
votes
0answers
70 views

Solve a PDE related to free boundary problem

I would like to solve the following system for my problem: $$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$ where $u=u(s,l): R\times R_+\to R$ is the unknown function ...
3
votes
1answer
212 views

Uniqueness of weak solutions of a heat equation

Let $M$ be a smooth compact closed manifold. Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of $$u_t - \Delta u - u = 0$$ $$u(0)=u(T)$$ satisfying $\int_M u(t) = 0$ for all $t$. Is ...
-2
votes
2answers
135 views

Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?
9
votes
1answer
267 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...