What a PDE Is

This book is about one equation and its relatives. The heat equation

says that the rate of change of a temperature at a point is proportional to how much the temperature there falls short of the average around it (2B.7 Fourier Series and the First Heat Equation). It is the simplest equation that smooths, the model for every diffusion in physics, chemistry, biology and finance, and, through Hamilton's insight, the model for the Ricci flow: $\partial_tg = -2\operatorname{Ric}(g)$ is a nonlinear heat equation for a Riemannian metric (11A.1 The Equation and Its First Solutions). Every analytic tool used on the Ricci flow is first a tool for the heat equation, and this book builds them in their simplest setting.

Before the tools, the vocabulary. This chapter says what a partial differential equation is, how equations are sorted into types, and what it means for one to be well posed. Then it proves two facts about the heat equation that hold for every parabolic equation and shape the whole subject. Parabolic scaling: diffusion over a distance $L$ takes a time proportional to $L^2$. Irreversibility: the heat equation can be solved forwards in time from any reasonable initial state, but not backwards.

By the end of this chapter you will be able to:

• say what the order of a PDE is, and whether it is linear, semilinear, quasilinear or fully nonlinear;
• classify second-order equations as elliptic, parabolic or hyperbolic from their symbol;
• state Hadamard's three conditions for well-posedness, and recognise an ill-posed problem;
• use parabolic scaling to estimate diffusion times, and find self-similar solutions of the heat equation;
• explain, with Fourier modes, why the backward heat equation is ill-posed and why deblurring an image is unstable.

1.1Diffusion takes the square of the distance

1.2What a PDE is

A partial differential equation is an equation relating an unknown function of several variables to its partial derivatives. Its order is the highest order of derivative that appears. Some of the equations this guide uses, with $u = u(x, t)$ or $u = u(x)$ for $x \in \mathbb{R}^n$:

equation	name	type
$u_t + b\cdot\nabla u = 0$	transport	linear, first order
$\Delta u = f$	Poisson (Laplace if $f = 0$)	linear, elliptic
$u_t = \Delta u$	heat	linear, parabolic
$u_{tt} = \Delta u$	wave	linear, hyperbolic
$u_t = \Delta u + u(1 - u)$	Fisher–KPP (6A.4 Maximum Principles)	semilinear, parabolic
$u_t = \Delta(u^m)$	porous medium (6A.7 Nonlinear Parabolic Equations)	quasilinear, parabolic (degenerate)
$\det D^2u = f$	Monge–Ampère (5A.6 Where This Track Leads)	fully nonlinear, elliptic
$\partial_tg = -2\operatorname{Ric}(g)$	Ricci flow (11A.1 The Equation and Its First Solutions)	quasilinear system, weakly parabolic

The words in the last column measure how nonlinear an equation is, according to where the nonlinearity sits. A second-order equation is

• linear if it has the form $\sum a^{ij}(x)\partial_i\partial_ju + \sum b^i(x)\partial_iu + c(x)u = f(x)$;
• semilinear if the coefficients of the highest derivatives depend only on $x$, and the nonlinearity is in the lower-order terms, as in Fisher–KPP;
• quasilinear if it is linear in the highest derivatives, with coefficients that may depend on $u$ and its lower derivatives: $\sum a^{ij}(x, u, \nabla u)\partial_i\partial_ju + \dots$;
• fully nonlinear if the highest derivatives enter nonlinearly, as in $\det D^2u$.

The more nonlinear, the harder. Semilinear equations behave much like linear ones on short time scales. Quasilinear equations can be handled by freezing the coefficients and solving a linear problem, which is the strategy of 6A.7 Nonlinear Parabolic Equations. The Ricci flow is quasilinear: written in coordinates, $\operatorname{Ric}$ involves second derivatives of $g$ linearly, with coefficients depending on $g$ itself.

1.3Elliptic, parabolic, hyperbolic

For a linear second-order operator $L u = \sum_{i,j}a^{ij}\partial_i\partial_ju + \text{lower order}$, with $(a^{ij})$ symmetric, the principal symbol is the quadratic form

It records what the operator does to a fast oscillation: if $u = e^{i\lambda x\cdot\xi}$ with $\lambda$ large, then $Lu \approx -\lambda^2\sigma(x, \xi)u$, since the second derivatives dominate. The operator is elliptic if $\sigma(x, \xi) > 0$ for every $\xi \neq 0$, that is, if the matrix $(a^{ij})$ is positive definite (2B.8 Calculus in Several Variables). The Laplacian is the model: $\sigma(\xi) = |\xi|^2$.

An evolution equation $u_t = Lu$ with $L$ elliptic is parabolic: the heat equation is the model. An equation $u_{tt} = Lu$ with $L$ elliptic is hyperbolic: the wave equation. The types behave completely differently. Elliptic equations describe equilibria and have smooth solutions; parabolic equations describe diffusion and smooth their data instantly; hyperbolic equations describe waves and carry singularities along at finite speed. This book is about the first two.

The symbol explains why. For a Fourier mode $u = \hat u(t)e^{2\pi ix\cdot\xi}$, the heat equation becomes the ODE $\hat u' = -4\pi^2|\xi|^2\hat u$ (4A.5 The Fourier Transform), so the mode decays at the rate $4\pi^2|\xi|^2$: the more it oscillates, the faster it dies. The positivity of the symbol is exactly what makes every oscillating mode decay. If $\sigma$ were negative in some direction, the corresponding modes would grow, and if $\sigma$ vanished in some direction, those modes would be left untouched.

1.4Well-posed problems

What does it mean to have solved a PDE? Jacques Hadamard proposed in 1902 that a problem for a PDE (an equation together with initial or boundary conditions) is well posed if

1. a solution exists;
2. it is unique;
3. it depends continuously on the data: small changes in the initial or boundary conditions produce small changes in the solution.

The third condition is the one that matters for applications. Data are measured, so they are never exact; a problem in which arbitrarily small errors in the data can produce large changes in the answer is useless for prediction, even if solutions exist and are unique. Which norms "small" refers to is part of the statement, and choosing them is part of the art.

Hadamard's own example of an ill-posed problem is the Laplace equation with initial data, as if it were an evolution equation. On $\mathbb{R}\times(0, \infty)$ consider $u_{xx} + u_{yy} = 0$ with $u(x, 0) = 0$ and $u_y(x, 0) = \frac1n\sin(nx)$. The solution is

As $n \to \infty$ the data tend to $0$ uniformly, but at $y = 1$ the solution has size $\frac{\sinh n}{n^2} \to \infty$. The zero solution has data arbitrarily close to that of a solution which is enormous a unit distance away. Elliptic equations are well posed with boundary conditions (the Dirichlet problem, 5A.4 Harmonic Functions and Conformal Mapping, 6A.2 Harmonic Functions), not with initial conditions.

1.5Parabolic scaling

Space and time scale differently: shrinking lengths by $\lambda$ shrinks times by $\lambda^2$. So the natural way to measure a region of space-time is by a length $r$ in space and $r^2$ in time, and the basic regions are parabolic cylinders $B_r(x) \times (t - r^2, t]$ (6A.6 Parabolic Regularity). The diffusion time estimate $t \approx L^2/2D$ is this proposition in physical units (Figure 1.1).

Scaling also finds solutions. Look for solutions that are unchanged by parabolic scaling up to a factor, of the form

In one dimension, substituting and writing $y = x/\sqrt t$ gives the ODE $F'' + \frac y2F' + aF = 0$ (Exercise 1.6). For $a = \frac12$ this is $\big(F' + \frac y2F\big)' = 0$, solved by $F(y) = e^{-y^2/4}$: we recover, up to a constant, the heat kernel $\frac{1}{\sqrt{4\pi t}}e^{-x^2/4t}$, the subject of 6A.3 The Heat Equation on ℝⁿ. The exponent $a = \frac12$ is forced by requiring the total heat $\int u\,dx$ to be conserved. For $a = 0$, the solution $F(y) = \int_0^ye^{-s^2/4}ds$ gives the error function profile: the temperature near the surface of a half-space suddenly heated, which Kelvin used to estimate the age of the Earth (6A.3 The Heat Equation on ℝⁿ).

1.6The backward heat equation is ill-posed

Forward in time, the heat equation is well posed in every reasonable sense (6A.3 The Heat Equation on ℝⁿ, 6A.4 Maximum Principles). Backward, it is not. On the circle $\mathbb{R}/\mathbb{Z}$, a solution is $u(x, t) = \sum c_ne^{-4\pi^2n^2t}e^{2\pi inx}$ (2B.7 Fourier Series and the First Heat Equation). Given the state at time $T$, recovering the state at time $0$ means multiplying the $n$-th Fourier coefficient by $e^{+4\pi^2n^2T}$.

A measurement of the present state, however accurate, contains errors at high frequency, and running the heat equation backwards multiplies them by factors that grow like $e^{cn^2}$. Worse, most states have no past at all: since every forward solution is smooth (indeed analytic, 5A.3 Residues and Fourier Transforms) at positive times, a final state that is not smooth cannot be the result of heat flow. The heat equation destroys information, at a rate that grows with frequency. That is the mathematical form of irreversibility, and it reappears as the monotonicity of entropy in 6A.10 Entropy, Information and Diffusion.

1.7History

Joseph Fourier's Théorie analytique de la chaleur (1822) derived the heat equation and solved it with series. Adolf Fick's law of diffusion (1855) put diffusing substances under the same equation. The classification of second-order equations into elliptic, parabolic and hyperbolic was settled in the nineteenth century, and the names are by analogy with conic sections: replacing derivatives by variables turns the model operators into polynomials whose level sets are an ellipse, a parabola and a hyperbola (Exercise 1.8). Jacques Hadamard formulated well-posedness in 1902 and gave his example of the ill-posed Cauchy problem for the Laplace equation; his book Lectures on Cauchy's Problem in Linear Partial Differential Equations (1923) developed it. The study of ill-posed problems and their regularisation became a field of its own in the 1960s, with Andrey Tikhonov's work.

1.8Exercises

Solution

(a) Second order, semilinear, parabolic. (b) Linear; elliptic for $x > 0$, hyperbolic for $x < 0$ (an equation of Tricomi type, which changes type across $x = 0$). (c) Second order, quasilinear: expanding, the coefficients of $\partial_i\partial_ju$ depend on $\nabla u$; parabolic, since the coefficient matrix $\frac{1}{\sqrt{1 + |p|^2}}\big(I - \frac{p\otimes p}{1 + |p|^2}\big)$ is positive definite. It is a close relative of the mean curvature flow of a graph (6A.8 Curve Shortening and the First Geometric Flows), which has an extra factor $\sqrt{1 + |\nabla u|^2}$ on the right. (d) Second order, fully nonlinear (a Monge–Ampère equation), elliptic at convex solutions. (e) First order, quasilinear (Burgers' equation).

Solution

(a) $L^2/2D$: $10^{-10}/4\times10^{-9} = 0.025$ s; $10^{-6}/4\times10^{-9} = 250$ s; $10^{-4}/4\times10^{-9} = 2.5 \times 10^4$ s, about $7$ hours; $1/4\times10^{-9} = 2.5 \times 10^8$ s, about $8$ years. (b) $10^{-4}/10^{-9} = 10^5$ s, a little over a day: convection from stirring (or from temperature differences) does the mixing, not diffusion. (c) $8 \times (3/2)^2 = 18$ minutes. The pan heats one side only, so the geometry is not symmetric; heat transfer at the surface is not instantaneous; meat contracts and loses water as it cooks.

Solution

(a) $u_t = -at^{-a-1}F - \frac12t^{-a-1}yF'$ and $u_{xx} = t^{-a-1}F''$. (b) $\int u\,dx = t^{-a}\int F(x/\sqrt t)\,dx = t^{\frac12 - a}\int F(y)\,dy$. (c) $a = \frac12$: $(F' + \frac y2F)' = 0$; the solution decaying at infinity has $F' + \frac y2F = 0$, so $F = Ce^{-y^2/4}$. $a = 0$: $F'' = -\frac y2F'$, so $F' = Ce^{-y^2/4}$ and $F = A + C\int_0^ye^{-s^2/4}\,ds$. (d) $F = Ce^{-r^2/4}$, the $n$-dimensional heat kernel $(4\pi t)^{-n/2}e^{-|x|^2/4t}$ when $C = (4\pi)^{-n/2}$.

Solution

We need $10^{-6}e^{4\pi^2n^2\cdot10^{-3}} \leq 10^{-2}$, i.e. $4\pi^2n^2 \times 10^{-3} \leq \ln10^4 \approx 9.21$, so $n^2 \leq 233$ and $n \leq 15$. Discarding all frequencies above $15$ trades the lost detail for stability: that is the simplest regularisation, spectral truncation.

Solution

(a) $\partial_tg_\lambda(t) = \lambda\cdot\frac1\lambda(\partial_tg)(t/\lambda) = -2\operatorname{Ric}(g(t/\lambda)) = -2\operatorname{Ric}(\lambda g(t/\lambda))$. (b) $\partial_tg = -2(n - 1)g_0$, and $\operatorname{Ric}(g(t)) = \operatorname{Ric}(g_0) = (n - 1)g_0$. (c) The sphere of radius $r$ is $g_\lambda(0)$ with $\lambda = r^2$, so its flow is $r^2g(t/r^2)$, which ends at $t = r^2T$.