Questions tagged [heat-equation]

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votes
1answer
50 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
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0answers
45 views

Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
5
votes
1answer
128 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 ...
3
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1answer
137 views

Ancient Heat equation and Liouville's theorem

I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation: \begin{equation} u_{t} = \...
2
votes
1answer
177 views

The solutions of the heat equation from $0$ datum

Consider the initial value problem $$ \partial_t u = \Delta u$$ $$ u(0,x) = 0$$ for the heat equation in $\mathbb R^n$, where $u: [0,T] \times \mathbb R^n \to \mathbb R$ is a smooth solution up to the ...
3
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1answer
350 views

Possible flaw in the proof of the Eells-Sampson theorem on harmonic maps in Nishikawa's book

Background: I am reading the book Variational Problems in Geometry by Seiki Nishikawa. The main purpose of this book is to prove the existence of harmonic maps $M\to N$ between two compact Riemannian ...
2
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0answers
91 views

Heat-Flow on continuous differential forms and the Feller peroperty

Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...
2
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0answers
136 views

Solution to Heat Equation By Projection

Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation $$ \partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x), $$ for some fixed $p\in C^2(\...
4
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0answers
75 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 ...
4
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1answer
283 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 ...
5
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3answers
202 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^...
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0answers
93 views

The complex roots of $\left(\Gamma(\kappa s)+2^{1-s} \pi^{-s}\cos\left(\frac{\pi s}{2} \right)\Gamma(s)\,\Gamma(\kappa(1-s))\right)$

Question: With $\kappa \in \mathbb{R}, \kappa \ge 1$, the function: $$\Upsilon(s,\kappa) = \Gamma(\kappa s)\zeta(s)+\Gamma\left(\kappa(1-s)\right) \zeta(1-s)$$ could be rewritten as: $$\Upsilon(s,\...
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0answers
60 views

A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow of a vector field

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\...
1
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3answers
278 views

One dimensional heat equation with boundary conditions

Consider the heat equation $$u_t = u_{xx}$$ for $t \ge 0$, $0 \le x \le L$, given boundary conditions $$u(0,t) = u(L,t) = f(t)$$ and an initial condition $$u(x,0) = g(x)$$ for some continuous ...
5
votes
1answer
209 views

Long time existence for heat flow in Corlette-Donaldson Theorem

I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14) https://arxiv.org/pdf/1402.4203.pdf For completeness, the statement is as follows. ...
1
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1answer
75 views

A question about positivity preserving property of semigroup of Laplacian

Sry this the second question from the following article, I am asking in this week. At page 6 (126), 3th line, of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say ...
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0answers
107 views

Diffusion equation solution using Laplace transform [closed]

Consider the operator $$ L=k\frac{\partial ^{2}}{\partial x^{2}}-\frac{\partial }{\partial t} $$ with domain $D(L)={u} \in \Bbb R \times [0,+\infty )$, initial value $u(x,0)=g(x), \forall x\in \Bbb R$...
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0answers
133 views

Convolution and approximate heat kernel

I have the following question: Let $A, B\in C^\infty((0,\infty)\times M\times M)$, where $M$ is a compact manifold. Define \begin{align} (A*B)(t,x,y) = \int_0^t\int_M A(t-s,x,z)B(s,z,y)dzds \end{align}...
2
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0answers
127 views

Feynman-Kac formula for domains with boundary

As far as I know (I am not an expert), any solution of the heat equation on $\mathbb{R}^n$ or on a closed Riemannian manifold with the given initial condition can be presented in terms of stochastic ...
0
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0answers
131 views

Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$. It is known that $S(t)$ ...
3
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2answers
177 views

Gaussian upper heat kernel bounds on closed Riemannian manifolds

Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists short time upper Gaussian heat kernel bounds of the following kind: $$ h(t, x, y) ...
2
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0answers
88 views

May the heat kernel of a connection Laplacian vanish?

Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...
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0answers
163 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D Laplace equation: $$\nabla^{2} T_w = 0$$ where $\nabla^{2}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [0,...
4
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0answers
103 views

large time behavior for the Neumann problem for the heat equation

I am interested in the large time asymptotic behavior of parabolic equations. For instance, let $\Omega$ be a regular bounded open subset of $\mathbb{R}^n$ . Let $g \in C(]0,+\infty[ \times \partial\...
3
votes
1answer
239 views

Bounded solution for parabolic equation

Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation $$u_t=\Delta u + a(x)u, \;\; (t,x)\...
16
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0answers
216 views

Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
3
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0answers
95 views

Ratio of solutions to two heat equations

Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
2
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0answers
135 views

Fundamental solution of parabolic PDE with variable coefficients

Let us consider the parabolic operator $$ \mathcal{L} = \partial_t - \nabla_x \cdot(a(x)\nabla_x) $$ over a bounded domain $\Omega\subset\mathbb{R}^d$. The coefficient matrix $a(x)$ is elliptic and ...
3
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0answers
90 views

The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....
2
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1answer
352 views

Heat kernel on Riemannian manifold

The idea to construct a heat kernel is first construct a parametric in a small neighbourhood. Then use a bump function to extend it. And do convolution iteratively. (Reference: Laplacian on a ...
4
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1answer
109 views

Quick question on the constants involved in heat kernel upper bounds

Let $M$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the Laplace-Beltrami operator on $M$. It is known that for small $t$, let's say, $0< t < t_0(M, g)$, the heat ...
13
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1answer
379 views

One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
3
votes
1answer
147 views

Gradient blowup for the 1-dim heat equation near an irregular boundary point

I'm trying to estimate the rate of boundary gradient blow-up for the 1-dim heat equation near an irregular boundary point. Let $b(t) := (1-t)^\alpha$, $\alpha < 1/2$. Let $u(t,x)$ solve the ...
0
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1answer
95 views

Analytical testcase for 2D/3D anisotropic Diffusion (Heat Kernel)

I want to verify and compare different Discretizations of the anisotropic diffusion equation in 2D / 3D image of my testsetting. I want to verify and compare different Discretizations of the ...
0
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1answer
283 views

Heat kernel and convergence

Let $(M_i,g_i,x_i)$ be a sequence of stochastically complete pointed Riemannanian manifolds ($x_i$ being the marked point on $M_i$) of injectivity radius uniformly bounded from below, Gromov-Haussdorf ...
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0answers
103 views

Heat equation, free boundary and dynamic programming

I have a dynamic programming problem with an underlying diffusion $$ d X_t = \mu \, dt + d b_t$$ where $b_t$ is a standard brownian motion. The HJB equation for the value function $v(x,t)$ I get is ...
5
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0answers
81 views

What is the generator of the heat semigroup on non-complete manifolds?

If $M$ is a complete Riemannian manifold, it possesses a unique self-adjoint positive operator $-\Delta$ on $L^2(M)$. If $M$ is not complete, though, it is known that the Laplace-Beltrami operator $-\...
4
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0answers
120 views

Davies' definition of elliptic operators in “Heat Kernels and Spectral Theory”

I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
5
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1answer
76 views

Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds

In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...
4
votes
1answer
423 views

$L^p$-norm under the heat flow

Let $(M, g)$ be a compact Riemannian manifold. Assume that $u_0$ is a positive smooth function on $M$ and let $u_t = e^{t \Delta} u_0$ be the solution to the heat equation on $(M, g)$ with initial ...
5
votes
2answers
291 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 $$ J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g. $$ ...
2
votes
1answer
228 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+\...
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0answers
54 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 ...
3
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0answers
178 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. ...
2
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0answers
69 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 $\...
3
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0answers
129 views

The uniqueness of fundamental solution on $\mathbb R^n$?

Let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\mathbb R^n$ which satisfies for all $y\in \mathbb R^n$: $(i) (\partial_t -\Delta_x )p_t(x,y)=0, \text{ }t>0, x\in \...
1
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1answer
232 views

Heat kernel asymptotic expansion on complete noncompact manifold

I was wondering if there is any reference for the asymptotic expansion of heat kernel on a complete noncompact manifold. That is, \begin{align} H(x,q,t) \sim \frac{e^{-\frac{d^2(q,x)}{4t}}}{(4\pi t)^{...
3
votes
1answer
186 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)...
3
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0answers
87 views

Reference request : maximal regularity for the heat equation on the torus

Fix $d\geq 1$ and $1<p,q<+\infty$. I am searching for a reference concerning the following result. There exists a constant $C=C(d,p,q)$ such that, for any $u\in\mathscr{C}^\infty(\mathbb{R}...
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0answers
97 views

Laplacian on squashed spheres

Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy $$ 1= \sum_{i=1}^n \omega_i |z_i|^2 $$ for fixed real numbers $\omega_i$? for ...