1.1 The Classical Heat Equation

In the most classical sense, the heat equation is the following partial differential equation on $\mathbb {R}^d \times \mathbb {R}$:

$$\left(\frac{\partial }{\partial t} - \sum \frac{\partial ^2}{\partial x_i^2}\right)f = 0.$$

This describes the dispersion of heat over time, where $f(x, t)$ is the temperature at position $x$ at time $t$. To simplify notation, we write

$$\Delta = - \sum \frac{\partial ^2}{\partial x_i^2}.$$

Green's strategy to solving such a PDE is to find a solution to the PDE with initial condition

$$f(x, 0) = \delta (x).$$

In this case, we can find it very explicitly to be

$$f(x, t) = \frac{1}{(4\pi t)^{d/2}} \exp \left(-\frac{|x|^2}{4t}\right).$$

We call this $H_t(x)$. Then given any bounded continuous function $f_0$, it is easy to see that the solution to the initial value problem

$$f(x, 0) = f_0(x)$$

is simply given by

$$f(x, t) = \int _{\mathbb {R}^d} H_t(x - y) f_0(y)\; \mathrm{d}y.$$

The fact that we are allowed to write $H_t(x - y)$ at all uses the fact that we are working on $\mathbb {R}^d$ and our system has translational symmetry. In general, if we work on an arbitrary manifold $M$, we would seek a function $H_t(x, y)$ on $M \times M \times (0, \infty )$ such that the solution to the initial value problem $f(x, 0) = f_0(x)$ is

$$f(x, t) = \int _M H_t(x, y) f_0(y)\; \mathrm{d}y.$$

The function $H_t(x, y)$ then satisfies

$$\left(\frac{\partial }{\partial t} + \Delta _x\right) H_t(x, y) = 0.$$

This $H_t(x, y)$ is also called the heat kernel, or fundamental solution, and we will mostly use these terms interchangeably. (It is also called a Green's function, but we will not use this name)

The heat kernel also shows up in a closely related problem. Suppose we wanted to solve instead

$$\left(\frac{\partial }{\partial t} - \Delta \right)f = F(x, t)$$

for some (bounded continuous) forcing term $F$. It turns out the solution can also be expressed in terms of $H$ as

$$f(x, t) = \int _M H_t(x, y) f_0(y)\; \mathrm{d}y + \int _0^t \mathrm{d}\tau \int _M H_{t - \tau }(x, y) F(y, \tau ) \; \mathrm{d}y,$$

which is again not difficult to check. The interpretation of this is that we can think of the forcing term as adding a new initial condition at each point $\tau$ in time. This is known as Duhamel's principle.

Especially from a physical perspective, it is interesting to note that the heat equation propagates information at infinite speed. In other words, for any $t > 0$, the value of $f(x, t)$ depends on the values of $f_0$ everywhere. This is in contrast with, for example, the wave equation, where information only propagates at finite speed. Nevertheless, in the limit $t \to 0$, the asymptotic behaviour is purely local, as the contribution of the points a finite distance away decays exponentially as $t \to 0$.

On a general Riemannian manifold, we can formulate the same problem, replacing $\Delta$ by the Laplace–Beltrami operator. The Laplace–Beltrami operator (or Laplacian for short) can in fact be defined for all $p$-forms. Write $\Omega ^* = \bigoplus _p \Omega ^p$. ¹ Then the exterior derivative defines a map

$$\mathrm{d}\colon \Omega ^* \to \Omega ^*.$$

There is an $L^2$ inner product on $p$-forms coming from the metric, and so $\mathrm{d}$ has a formal adjoint $\mathrm{d}^*\colon \Omega ^* \to \Omega ^*$ that lowers degree by $1$. The Laplace–Beltrami operator is given by

$$\Delta = (\mathrm{d}+ \mathrm{d}^*)^2 = \mathrm{d}^* \mathrm{d}+ \mathrm{d}\mathrm{d}^* \colon \Omega ^p \to \Omega ^p.$$

In the case of $p = 0$, the Laplace–Beltrami operator is simply given by $\mathrm{d}^* \mathrm{d}$, and in local coordinates with metric $g$, we can write this as

$$\Delta f = \frac{1}{\sqrt{\det g}} \partial _i (\sqrt{\det g}\, g^{ij} \partial _j f).$$