Fundamental solution to the heat equation, given boundary values
In the
mathematical study of
heat conduction and
diffusion , a heat kernel is the
fundamental solution to the
heat equation on a specified domain with appropriate
boundary conditions . It is also one of the main tools in the study of the
spectrum of the
Laplace operator , and is thus of some auxiliary importance throughout
mathematical physics . The heat kernel represents the evolution of
temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0 .
Fundamental solution of the one-dimensional heat equation. Red: time course of
Φ
(
x
,
t
)
{\displaystyle \Phi (x,t)}
. Blue: time courses of
Φ
(
x
0
,
t
)
{\displaystyle \Phi (x_{0},t)}
for two selected points.
Interactive version.
The most well-known heat kernel is the heat kernel of d -dimensional
Euclidean space R d , which has the form of a time-varying
Gaussian function ,
K
(
t
,
x
,
y
)
=
exp
(
t
Δ
)
(
x
,
y
)
=
1
(
4
π
t
)
d
/
2
e
−
‖
x
−
y
‖
2
/
4
t
{\displaystyle K(t,x,y)=\exp \left(t\Delta \right)(x,y)={\frac {1}{\left(4\pi t\right)^{d/2}}}e^{-\|x-y\|^{2}/4t}}
which is defined for all
x
,
y
∈
R
d
{\displaystyle x,y\in \mathbb {R} ^{d}}
and
t
>
0
{\displaystyle t>0}
. This solves the heat equation
{
∂
K
∂
t
(
t
,
x
,
y
)
=
Δ
x
K
(
t
,
x
,
y
)
lim
t
→
0
K
(
t
,
x
,
y
)
=
δ
(
x
−
y
)
=
δ
x
(
y
)
{\displaystyle \left\{{\begin{aligned}&{\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)\\&\lim _{t\to 0}K(t,x,y)=\delta (x-y)=\delta _{x}(y)\end{aligned}}\right.}
where
δ is a
Dirac delta distribution and the limit is taken in the sense of
distributions , that is, for every smooth function
ϕ of
compact support , we have
lim
t
→
0
∫
R
d
K
(
t
,
x
,
y
)
ϕ
(
y
)
d
y
=
ϕ
(
x
)
.
{\displaystyle \lim _{t\to 0}\int _{\mathbb {R} ^{d}}K(t,x,y)\phi (y)\,dy=\phi (x).}
On a more general domain Ω in R d , such an explicit formula is not generally possible. The next simplest cases of a disc or square involve, respectively,
Bessel functions and
Jacobi theta functions . Nevertheless, the heat kernel still exists and is
smooth for t > 0 on arbitrary domains and indeed on any
Riemannian manifold
with boundary , provided the boundary is sufficiently regular. More precisely, in these more general domains, the heat kernel the solution of the initial boundary value problem
{
∂
K
∂
t
(
t
,
x
,
y
)
=
Δ
x
K
(
t
,
x
,
y
)
for all
t
>
0
and
x
,
y
∈
Ω
lim
t
→
0
K
(
t
,
x
,
y
)
=
δ
x
(
y
)
for all
x
,
y
∈
Ω
K
(
t
,
x
,
y
)
=
0
,
x
∈
∂
Ω
or
y
∈
∂
Ω
.
{\displaystyle \left\{{\begin{aligned}&{\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)&\quad {\text{ for all }}t>0{\text{ and }}x,y\in \Omega \\[6pt]&\lim _{t\to 0}K(t,x,y)=\delta _{x}(y)&{\text{ for all }}x,y\in \Omega \qquad \qquad \,\\[6pt]&K(t,x,y)=0,&x\in \partial \Omega {\text{ or }}y\in \partial \Omega \qquad \,\,\,\,.\end{aligned}}\right.}
It is not difficult to derive a formal expression for the heat kernel on an arbitrary domain. Consider the Dirichlet problem in a connected domain (or manifold with boundary) U . Let λ n be the
eigenvalues for the Dirichlet problem of the Laplacian
{
Δ
ϕ
+
λ
ϕ
=
0
in
U
ϕ
=
0
on
∂
U
.
{\displaystyle \left\{{\begin{array}{ll}\Delta \phi +\lambda \phi =0&{\text{in }}U\\\phi =0&{\text{on }}\ \partial U.\end{array}}\right.}
Let
ϕ n denote the associated
eigenfunctions , normalized to be orthonormal in
L 2 (U ). The inverse Dirichlet Laplacian
Δ−1 is a
compact and
selfadjoint operator , and so the
spectral theorem implies that the eigenvalues of
Δ satisfy
0
<
λ
1
≤
λ
2
≤
λ
3
≤
⋯
,
λ
n
→
∞
.
{\displaystyle 0<\lambda _{1}\leq \lambda _{2}\leq \lambda _{3}\leq \cdots ,\quad \lambda _{n}\to \infty .}
The heat kernel has the following expression:
K
(
t
,
x
,
y
)
=
∑
n
=
0
∞
e
−
λ
n
t
ϕ
n
(
x
)
ϕ
n
(
y
)
.
{\displaystyle K(t,x,y)=\sum _{n=0}^{\infty }e^{-\lambda _{n}t}\phi _{n}(x)\phi _{n}(y).}
(1 )
Formally differentiating the series under the sign of the summation shows that this should satisfy the heat equation. However, convergence and regularity of the series are quite delicate.
The heat kernel is also sometimes identified with the associated
integral transform , defined for compactly supported smooth ϕ by
T
ϕ
=
∫
Ω
K
(
t
,
x
,
y
)
ϕ
(
y
)
d
y
.
{\displaystyle T\phi =\int _{\Omega }K(t,x,y)\phi (y)\,dy.}
The
spectral mapping theorem gives a representation of
T in the form
T
=
e
t
Δ
.
{\displaystyle T=e^{t\Delta }.}
There are several geometric results on heat kernels on manifolds; say, short-time asymptotics, long-time asymptotics, and upper/lower bounds of Gaussian type.
See also
References
Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators , Berlin, New York:
Springer-Verlag
Chavel, Isaac (1984), Eigenvalues in Riemannian geometry , Pure and Applied Mathematics, vol. 115, Boston, MA:
Academic Press ,
ISBN
978-0-12-170640-1 ,
MR
0768584 .
Evans, Lawrence C. (1998), Partial differential equations , Providence, R.I.:
American Mathematical Society ,
ISBN
978-0-8218-0772-9
Gilkey, Peter B. (1994),
Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem ,
ISBN
978-0-8493-7874-4
Grigor'yan, Alexander (2009),
Heat kernel and analysis on manifolds , AMS/IP Studies in Advanced Mathematics, vol. 47, Providence, R.I.:
American Mathematical Society ,
ISBN
978-0-8218-4935-4 ,
MR
2569498
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