Questions tagged [heat-equation]
The heat-equation tag has no usage guidance.
148
questions
2
votes
1answer
50 views
Explicit solution of a free boundary problem for heat equation
Consider the free boundary problem
$$
\min\{u_t - u_{xx} -1, u \} = 0 \qquad \text{ in } (0,T)\times (-1,1) \\
u(0,\cdot) = 0 \qquad \text{ in } (-1,1)\\
u(\cdot, -1) = u(\cdot, 1) = 0 \qquad \...
1
vote
0answers
45 views
Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)
I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it:
...
5
votes
1answer
128 views
Backward uniqueness for a heat equation with a drift
Consider heat equation with a drift (=reaction-diffusion equation)
$$
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1]
$$
with periodic or ...
3
votes
1answer
137 views
Ancient Heat equation and Liouville's theorem
I encounter a difficulty when proving the bounded solution of ancient heat equation implying constant function. Suppose $u(t,x)$ is the solution of ancient heat equation:
\begin{equation}
u_{t} = \...
2
votes
1answer
177 views
The solutions of the heat equation from $0$ datum
Consider the initial value problem
$$ \partial_t u = \Delta u$$
$$ u(0,x) = 0$$
for the heat equation in $\mathbb R^n$, where $u: [0,T] \times \mathbb R^n \to \mathbb R$ is a smooth solution up to the ...
3
votes
1answer
350 views
Possible flaw in the proof of the Eells-Sampson theorem on harmonic maps in Nishikawa's book
Background:
I am reading the book Variational Problems in Geometry by Seiki Nishikawa. The main purpose of this book is to prove the existence of harmonic maps $M\to N$ between two compact Riemannian ...
2
votes
0answers
91 views
Heat-Flow on continuous differential forms and the Feller peroperty
Let (M,g) be a complete Riemannian manifold. It is well known that the Laplace operator is essentially self-adjoint on $C^\infty_c(M)$. This extends to the (de Rahm) Laplace operator on forms. Thus in ...
2
votes
0answers
136 views
Solution to Heat Equation By Projection
Let $X\subseteq C(\mathbb{R}\times [0,\infty))$ be the collection of all solutions to the IV heat equation
$$
\partial_t u(t,x) = \partial^2 u(t,x) \qquad u(0,x)=p(0,x),
$$
for some fixed $p\in C^2(\...
4
votes
0answers
75 views
Biharmonic heat flow on compact manifolds
Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...
4
votes
1answer
283 views
Lack of exponential $L^2_{t,x}$ decay for a heat equation with growing coefficients
Edit: I have changed the nature of the question, but in order to have a better idea of what I can expect for the original problem (see below).
Given $T>0$ and $n \in \bf Z$, consider the following ...
5
votes
3answers
202 views
Reference request: Long-term behaviour of the heat equation for bounded initial data
Let us consider the heat equation
\begin{align*}
\frac{\partial}{\partial t}u(t,x) & = \Delta u(t,x), \\
u(0,x) & = f(x)
\end{align*}
on the whole space $\mathbb{R^d}$. If $f \in L^p := L^...
1
vote
0answers
93 views
The complex roots of $\left(\Gamma(\kappa s)+2^{1-s} \pi^{-s}\cos\left(\frac{\pi s}{2} \right)\Gamma(s)\,\Gamma(\kappa(1-s))\right)$
Question:
With $\kappa \in \mathbb{R}, \kappa \ge 1$, the function:
$$\Upsilon(s,\kappa) = \Gamma(\kappa s)\zeta(s)+\Gamma\left(\kappa(1-s)\right) \zeta(1-s)$$
could be rewritten as:
$$\Upsilon(s,\...
0
votes
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
vote
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
vote
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 ...
1
vote
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 $u(x,0)=g(x), \forall x\in \Bbb R$...
2
votes
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
votes
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
votes
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
votes
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:
$$
h(t, x, y) ...
2
votes
0answers
88 views
May the heat kernel of a connection Laplacian vanish?
Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...
0
votes
0answers
163 views
3D Homogenous Laplace equation with integral boundary conditions
I have the 3D Laplace equation:
$$\nabla^{2} T_w = 0$$
where $\nabla^{2}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [0,...
4
votes
0answers
103 views
large time behavior for the Neumann problem for the heat equation
I am interested in the large time asymptotic behavior of parabolic equations. For instance, let $\Omega$ be a regular bounded open subset of $\mathbb{R}^n$ . Let $g \in C(]0,+\infty[ \times \partial\...
3
votes
1answer
239 views
Bounded solution for parabolic equation
Let $\Omega_T=(0,T) \times \Omega$, where $\Omega$ a bounded smooth domain of $\mathbb{R}^n$ and $T>0$. Let $a\in L^\infty(\Omega)$ and consider the heat equation
$$u_t=\Delta u + a(x)u, \;\; (t,x)\...
16
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
votes
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+\...
1
vote
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
votes
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
votes
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
votes
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
vote
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
votes
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}...
1
vote
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 ...
