The heat-equation tag has no wiki summary.
2
votes
0answers
112 views
Heat kernel and Wiener measure
A theorem by Barry Simon says that for arbitrary open sets $\Omega\subset \mathbb{R}^n$, we have $$[\exp(t\Delta_{\Omega}^D)](x,y) = \mu_{x,y,t}\lbrace \omega \text{ } \vert \text{ } \omega(s) \in ...
0
votes
0answers
53 views
Heat equation with heterogeneous heat conduction
I'm trying to discretize and a heat conduction/diffusion problem using finite differences and I was wondering how to use a discrete heat conduction coefficient defined per cell (instead of per ...
5
votes
1answer
800 views
Mathematical study of Mpemba effect?
It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
8
votes
1answer
299 views
Atiyah-Singer for pseudodifferential operators via heat kernel?
The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...
1
vote
0answers
57 views
monotone parabolic systems, convex variational structure and Legendre transform
The context:
for my research I am currently looking at parabolic systems of the type
$$
\left\{
\begin{array}{ll}
\partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\
u=0 & ...
3
votes
0answers
79 views
$L^2$ bounds for the gradient of subsolutions to parabolic equation
Suppose we have the differential inequality
$$
|\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|)
$$
in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we ...
4
votes
1answer
73 views
Examples of optimal ultracontractivity estimates for a Markovian semigroup $T_t$ that do not depend polynomialy on $t$
Let $(X,\mu)$ be a measure space and $T_t : L_2(\mu) \to L_2(\mu)$ for $t \geq 0$ a symmetric Markovian semigroup. Local ultracontractivity estimates of the form:
$$
\| T_t : L_p(\mu) \to L_q(\mu)\| ...
1
vote
0answers
99 views
localization of the $L^p$ variation for heat equation
I'm struggling with yet another question for the classical heat equation in the whole space $R^d$. This question seems basic at first sight, but I think it is nontrivial in the end so here it is.
The ...
2
votes
1answer
254 views
Mellin transform between heat kernel and zeta-function
For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...
1
vote
0answers
82 views
Gaussian upper bounds on Heat kernel
Let $M$ be a complete Riemannian manifold with Ricci curvature bounded below and let $k(t, x, y)$ be the heat kernel of the Laplace-Beltrami operator.
It seems to be standard that for any compact ...
3
votes
1answer
192 views
Heat Equation on $[0,T] \times \mathbb{R}^n$
I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.
Do you ...
2
votes
1answer
224 views
Geodesics and harmonic map heat flow
I believe the answer to my question is well-known, but as I do not know too much about harmonic map flows, here it goes:
Let $M$ be a complete Riemannian manifold. For a curves $c: [0,1] ...
2
votes
1answer
205 views
The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold
This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne.
Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ ...
3
votes
0answers
60 views
Heat Kernel Asymptotics with low regularity
Let $M$ be a smooth manifold with Riemannian metric $g$, which is not smooth but only continuous.
Question: Is there still an asymptotic expansion of the heat kernel of the form
$$ p_t(x, y) \sim (4 ...
22
votes
3answers
877 views
“Wild” solutions of the heat equation: how to graph them?
It has long been known that the Cauchy initial-value problem for the
classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't
have unique solutions, without additional assumptions. In ...
5
votes
0answers
212 views
Heat Kernel Asymptotics on Manifold with Boundary
This is crosspost from math.stackexchange http://math.stackexchange.com/questions/311213/heat-kernel-asymptotics-on-manifold-with-boundary where the question did not yield any answer
On a closed ...
5
votes
1answer
203 views
Asymptotic Expansion of the Schrödinger kernel?
My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough what my interest ...
3
votes
0answers
136 views
Heat kernel proof of Poincaré Hopf
I am familiar with Witten's proof of the Morse inequalities by semiclassical analysis. Hey uses semiclassical expansions of the first eigenfunctions to construct the Morse complex from it, which ...
1
vote
1answer
702 views
Solution of Heat equation with Neumann BC in an arbitrary domain
Consider the heat equation $u_t=\Delta u$ with Neumann boundary condition and initial condition $u(x,0)=u^0(x)$ in a bounded domain $\Omega$ with smooth boundary.
Is this true:
Any solution ...
4
votes
3answers
361 views
Reference for estimation gaussian of the heat kernel
Let $(M,g^{TM})$ a Riemannian manifold of dimension $n$ and $\Delta$ the Laplace–Beltrami operator. I would like to find a reference (analytic or probabilistic) for the following classic result.
...
5
votes
1answer
974 views
Non-uniqueness of solutions of the heat equation
For the heat equation $(\partial_t-\partial_x^2)f(t,x)=0$ defined on $[0,T)\times(-\infty,\infty)$, to obtain uniqueness of the initial value problem, usually it is required to limit the growth of the ...
4
votes
2answers
494 views
How do you solve linear systems whose solutions decay exponentially?
Consider the heat equation
$$\dot{u} = \Delta u$$
with initial conditions
$$u_0 = \delta(x)$$
for some point $x$ in the domain $\Omega$ of the problem. If $\Omega$ is $\mathbb{R}^n$, then this ...
26
votes
1answer
2k views
Unconditional nonexistence for the heat equation with rapidly growing data?
Consider the initial value problem
$$ \partial_t u = \partial_{xx} u$$
$$ u(0,x) = u_0(x)$$
for the heat equation in one dimension, where $u_0: {\bf R} \to {\bf R}$ is a smooth initial datum and $u: ...
