Questions tagged [heat-equation]
The heat-equation tag has no usage guidance.
2
votes
1answer
122 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
$$u_t=\Delta u + a(x)u, \;\; (t,x)\...
13
votes
0answers
171 views
Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?
Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...
1
vote
0answers
102 views
Solution of nonlinear heat equation decreases in time
Let $f\colon \mathbb{R} \to \mathbb{R}$ be a smooth decreasing function which is non-negative and bounded above, and consider the heat equation on a smooth bounded domain
$$u_t - \Delta u = f(u)$$
...
3
votes
0answers
83 views
Ratio of solutions to two heat equations
Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
1
vote
0answers
49 views
Fundamental solution of parabolic PDE with variable coefficients
Let us consider the parabolic operator
$$
\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
votes
0answers
66 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....
3
votes
1answer
185 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 ...
3
votes
1answer
82 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 ...
11
votes
1answer
332 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
126 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
votes
1answer
69 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
votes
1answer
180 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 ...
1
vote
0answers
79 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$$
where $b_t$ is a standard brownian motion.
The HJB equation for the value function $v(x,t)$ I get is ...
5
votes
0answers
68 views
What is the generator of the heat semigroup on non-complete manifolds?
If $M$ is a complete Riemannian manifold, it possesses a unique self-adjoint positive operator $-\Delta$ on $L^2(M)$. If $M$ is not complete, though, it is known that the Laplace-Beltrami operator $-\...
4
votes
0answers
100 views
Davies' definition of elliptic operators in “Heat Kernels and Spectral Theory”
I am trying to find my way through Davies' book, and one of the difficult points is his choice of what "elliptic operator" means in his text. This is the first time that I encounter some of the ...
6
votes
1answer
64 views
Gaussian bounds for the heat kernel of regular domains in Riemannian manifolds
In "Heat Kernels and Spectral Theory" Davies constructs upper and lower bounds for the kernels associated to Dirichlet elliptic operators on regular domains of $\mathbb R^n$. Has anybody done the same ...
3
votes
1answer
381 views
$L^p$-norm under the heat flow
Let $(M, g)$ be a compact Riemannian manifold.
Assume that $u_0$ is a positive smooth function on $M$ and let $u_t = e^{t \Delta} u_0$ be the solution to the heat equation on $(M, g)$ with initial ...
5
votes
2answers
274 views
Functional decaying under the heat flow (?)
Let $(M, g)$ be a compact Riemannian manifold and let $a$, $p$ be two real numbers greater than $1$.
For any positive function $v$, I set
$$
J(v) = \int_M \left|\nabla(v^a)\right|^p d\mu^g.
$$
...
1
vote
1answer
99 views
schauder regularity heat equation
Let $m \in \mathbb{N}\setminus \{0,1\}$, $\alpha \in ]0,1[$.
Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{m,\alpha}$.
It is known that if $f \in C^{\frac{m-2+\alpha}{2},m-2+\...
1
vote
0answers
52 views
Parabolic (heat) PDE Green's function spatial asymptote at infinity
Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its ...
1
vote
0answers
135 views
Critical spaces and energy estimate in NS equation [closed]
There is a ‘rescaling transformation’ that is particularly significant for
the Navier–Stokes equations when they are posed on the whole space, but is
also important in the local regularity theory.
...
1
vote
0answers
50 views
Inhomogeneous heat kernel estimates
I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation
$$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$
where $\...
3
votes
0answers
86 views
The uniqueness of fundamental solution on $\mathbb R^n$?
Let $p_t(x,y)$ be a non-negative fundamental solution to the heat equation on $\mathbb R^n$ which satisfies for all $y\in \mathbb R^n$:
$(i) (\partial_t -\Delta_x )p_t(x,y)=0, \text{ }t>0, x\in \...
1
vote
1answer
104 views
Heat kernel asymptotic expansion on complete noncompact manifold
I was wondering if there is any reference for the asymptotic expansion of heat kernel on a complete noncompact manifold. That is,
\begin{align}
H(x,q,t) \sim \frac{e^{-\frac{d^2(q,x)}{4t}}}{(4\pi t)^{...
3
votes
1answer
147 views
Parabolic Regularity with Neumann B.C
Consider the parabolic problem in the cylinder of base $B$, the unit ball,
$$
\partial_t u -\text{div}\left( A(x) D u +F(t,x)\right)=0 \text{ in } (0,T)\times B,
$$
with $(ADu +F)\cdot \nu=0$ on $(0,T)...
3
votes
0answers
65 views
Reference request : maximal regularity for the heat equation on the torus
Fix $d\geq 1$ and $1<p,q<+\infty$. I am searching for a reference concerning the following result.
There exists a constant $C=C(d,p,q)$ such that, for any
$u\in\mathscr{C}^\infty(\mathbb{R}...
1
vote
0answers
91 views
Laplacian on squashed spheres
Is anything known about the Laplacian on squashed spheres $S^{2n-1}_\omega$, where the ambient $C^n$ coordinates satisfy
$$ 1= \sum_{i=1}^n \omega_i |z_i|^2 $$
for fixed real numbers $\omega_i$? for ...
2
votes
0answers
125 views
Is Varadhan's formula valid for all pairs of points?
Most formulations of Varadhan's formula
$$\lim _{t \to 0_+} 4t \log p_t(x,y) = -d(x,y)^2$$
that I have encountered do not specify where $(x,y)$ lives, so until today I imagined that $(x,y) \in M \...
4
votes
2answers
184 views
Maximum principle for heat equation, low regularity case
I meant to assign to my class the following homework problem:
If $u\in C^2((0,T)\times \Omega) \cap C^0([0,T]\times\bar{\Omega})$ where $\Omega$ is an open, bounded domain, is such that $\partial_t ...
1
vote
1answer
80 views
Decay time to constant function of heat kernel on 2-sphere
Let us consider solving the heat equation on the sphere given a delta function as initial data.
$$(\partial_t - \Delta)K(x,y;t) = 0 $$
$$K(x,y;0) = \delta(x,y)$$
One would expect that for large ...
2
votes
0answers
103 views
hyperbolic “Green function” on a product of upper half-planes
Let $\Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2)$ be the hyperbolic Laplacian acting on functions of $\mathfrak{h}$ (the Poincare upper half-plane) and consider its resolvent
$$
R(s)=(...
6
votes
0answers
193 views
How to interpret heat kernel at unit time on a Riemann surface?
Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation
$$
(\partial_{...
4
votes
2answers
390 views
Gaussian distribution, maximum entropy and the heat equation
I have asked this question on MathSE, but I got no replies, so I thought of trying here.
Consider the Gaussian distribution on $\mathbb{R}$ with mean $m$ and variance $t=\sigma^2$. This has the ...
3
votes
1answer
170 views
Local upper estimates for Neumann heat kernels
I have a question about Neumann heat kernels and its estimates.
Let $D$ be a domain of $\mathbb{R}^d$. We define the Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^{2}(D)$ as follows:
\begin{align*}...
0
votes
1answer
97 views
Proof of the Davies-Gaffney estimate in elliptic pdes?
I'd like a reference to a proof of the Davies-Gaffney estimate; which is an off-diagonal decay result. See for instance assumption H2 in the paper "Hardy Spaces associated to non-negative self-adjoint ...
1
vote
0answers
53 views
Heat equation with source term in $L^1$
To simplify, let us work on $Q_T:=[0,T]\times\mathbb{T}^N$ where $\mathbb{T}^N$ is the $N$-th dimensionnal torus.
Consider $(S_n)_n$ a sequence of $L^1(Q_T)$ and $(z_n)_n$ the sequence of solutions ...
2
votes
0answers
94 views
explicit formulae of heat kernel on graphs
I have just discovered this article about heat kernels on graphs. It has been written by a respected theoretical physicist, but seemingly never made it into a peer-reviewed journal. On the other hand, ...
5
votes
1answer
272 views
Heat Equation with an integral boundary condition
I have been struggling with following Heat equation IBVP,
\begin{equation}
\frac{\partial v\left(x, t\right)}{\partial t} = \alpha \frac{\partial^2 v\left(x, t\right)}{\partial x^2}, \quad t \in \left(...
7
votes
2answers
297 views
Intuition for the Drift Term of the Laplace-Beltrami Operator
In coordinates, the Laplace-Beltrami operator on a Riemannian manifold $(M,g)$ can be written as:
$$
\Delta_g = g^{ij}\partial_{ij} - g^{jk}\Gamma^\ell_{jk}\partial_\ell
$$
The second term:
$$
\mu^\...
1
vote
1answer
163 views
Gevrey estimate of derivatives
Let $\phi$ be the $C^\infty$ function defined on $\mathbb R_+^*$ by $e^{-t^{-2}}$ and by $0$ on $\mathbb R_-$.
Question: I think that there exists $\rho>0$ such that
$$
\forall t\in \mathbb R,\...
1
vote
0answers
55 views
Reference: Varadhan's lemma for Finsler Geometry?
Is there a version of Varadhan's lemma for heat-kernels on Finsler manifolds? I expect this to exist but I cannot seem to find any papers on the topic. References would be greatly appreciated.
4
votes
2answers
277 views
Heat equation close to the steady state
Let $u(t, x)$ be the unique solution of the heat equation on the unit interval with Dirichlet boundary conditions and initial data $u_0$:
$$
\left\{
\begin{array}{l}
\partial_t u(t, x) = \partial_x^2 ...
1
vote
0answers
55 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 ...
8
votes
0answers
184 views
Basis for $L^2(\mathbb{R})$ that Solves the Heat Equation
This is a less-than-serious question that I asked on math.SE, but I suspect it is slightly more appropriate to ask it here. Consider the heat equation $$
u_t = \frac12 u_{xx}
$$ On $\mathbb{T}$ with ...
3
votes
0answers
85 views
Are heat kernels on metric measure spaces continuous?
Let $(M,d)$ be a separable, complete, compact metric space and $\mu$ a Radon measure with full support on it. Let $\mathcal{E}$ be a regular strongly local Dirichlet form on $L^2(M)$. There exists an ...
2
votes
1answer
183 views
Singularity of the heat kernel
The heat kernel in one dimension for the real line is given by the usual gaussian density function:
$$g(t,x,y)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{(x-y)^2}{2t}}\, .$$
In particular, by differentiating ...
4
votes
1answer
249 views
heat kernel on closed manifolds - error in Chavel's book?
first of all, I am not sure if this question fits here. I asked this question on math.stackexchange also but didn't get an answer so far.
In Isaac Chavel's book Eigenvalues in Riemannian Geometry, ...
8
votes
1answer
312 views
Long-time decay of heat kernel on compact manifolds
Let $M$ be a compact Riemannian manifold and let $V \in C^\infty(M)$. Consider the operator $\Delta + V$ and let $p_t(x, y)$ be the corresponding heat kernel. If $\Delta + V$ is a positive operator ...
4
votes
1answer
111 views
Geodesic-like curves stemming from the heat kernel on a manifold
Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>...
4
votes
0answers
88 views
Decay of frequencies of solution of the heat equation with a potential
Let $I_k = \{f \in L^2(R^n); supp(\hat{f}) \subset B(0,k) \}$.
Let $\Pi_k$ the orthogonal projection on $I_k$.
Let $a(t,x)$ a regular bounded potential.
Let $f$ the solution in $L^2$ to the Cauchy ...
