# PDE’s Greatest Hits

When I was an undergraduate I took a course in Galois theory that was something like the “greatest hits of Galois theory [accessible to an undergraduate]”. Thus we proved the insolubility of the quintic, classified finite fields up to isomorphism, and proved that one cannot square a circle. Recently I’ve been thinking about what would go in a similar course in PDE, which seems like a very natural field in which one could hold a “greatest hits” course for a few reasons. Since PDE isn’t part of the standard curriculum, if one ends up not covering a particular topic because the students are more interested in another topic, it will not harm them when they take future courses. Moreover, the field of PDE is much more about techniques than theorems, and the techniques can be taught using any particular equations that are physically or computationally interesting. Students could also, in lieu of a final exam, give a 20-minute talk in the last few weeks of the course about an application they found particularly interesting.

I’d be interested to hear what sort of “greatest hits” your field has, but here’s my thoughts on the greatest hits of PDE.

There would probably have to be a foundational section of the course, which discusses the following topics, that are indispensable in the study of PDE:

The Laplace equation. We start by introducing the $L^p$ norms informally: we discuss no duality theory and appeal to no measure theory, but only state the Hoelder and Minkowski inequalities, and take completeness as a black box. We similarly define the $C^r$ norms.

We now introduce the Dirichlet energy $I[u] = ||\nabla u||_{L^2}^2$ in several different forms: a quantity that is minimized by a chemical system in equilibrium, a term in the Mumford-Shah energy from image processing, and a linear approximation to the Lagrangian action $||\sqrt{1 + |\nabla u|^2}||_{L^1}$ for minimal graphs. The case of minimal graphs would be particularly fun to teach, as one can bring in bubble wands and try to predict the shapes of soap films, which locally are minimal graphs. (This activity was suggested in a talk of Jenny Harrison.) We then introduce the notions of Lagrangian and Laplacian, and deduce that

Let $u \in C^2$. Then $I[u]$ is minimal subject to Dirichlet boundary data iff u solves the Laplace equation.

For the sake of later discussion, we also introduce the heat equation as the gradient flow of the Dirichlet energy.

So now we have motivated the Laplace equation. We argue that $I[u]$ is invariant under the rotation group $SO(n)$, which gives the formula for a Newtonian potential almost immediately. The Dirichlet energy also gives an easy proof of uniqueness for the Laplace equation. We then introduce the notion of convolution, motivated by signal-processing filters. Convolution against Gaussians gives an easy proof that harmonic functions are smooth and convolution against the Newtonian potential solves the Laplace equation. (The reason I want to use Gaussians here is that constructing smooth functions of compact support is an annoying technical argument that not all students may be comfortable with.) We conclude by proving the maximum modulus principle, which implies the fundamental theorem of algebra.

Baby’s first harmonic analysis course. We start with a history lesson: In 1798, Joseph Fourier joins Napoleon Bonaparte on his conquest of Egypt. Twenty-four years later, Fourier releases his controversial treatise on the heat equation, in which he uses some seemingly dubious methods to approximate functions by trigonometric polynomials. Thus it is our job to make right what Dr. Fourier got wrong.

So we introduce the space $C^\infty(T^n)$ of smooth functions on a torus, and argue that we can approximate them using trigonometric polynomials. Here we take the Stone-Weierstrass theorem as a black box. Taking dual spaces, we introduce the Dirac delta function as the unit of convolution and then define periodic distributions. Since this is a course for undergraduates, it is crucial that this step can be carried out without introducing the notion of a Fréchet space, as one just needs to define covectors on $C^\infty(T^n)$ to be those linear maps which satisfy the suitable inequalities. Thus we solve the heat equation subject to smooth initial data and periodic boundary data. We also observe that the Laplace equation has no solutions subject to periodic boundary data except the zero solution.

But, contra to Nietzsche, time is not a flat circle and so we want to solve these equations on euclidean space. So now we introduce the Schwartz space, motivated by our use of $C^\infty(T^n)$, and the Fourier transform on Schwartz space. We then prove the Fourier inversion formula and introduce the notion of a tempered distribution.

At this point, we hit upon a fundamental theorem:

If a linear PDE is invariant under the group $\mathbb R^n$, then it is diagonalized by the Fourier transform.

As a corollary, the heat equation is immediately solved. We also give another derivation of the Newtonian potential, using invariance under $\mathbb R^n$ rather than $SO(n)$ this time.

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At this point we’re probably a third of the way through the semester, and have touched upon several key themes: the use of the Fourier transform to solve equations, the importance of the symmetry group, and the importance of estimates. Now it is time for the students to pick topics for us to cover for the rest of the semester, based on their interests, since we certainly will not have time to study them all. We pick from:

The wave equation and Einstein’s theory of special relativity. Taking the Maxwell equation as a given, we deduce the curl-of-curl theorem and deduce that the Maxwell equation is actually a system of wave equations. This allows us to carry out Maxwell’s argument that light is an electromagnetic wave.

We now digress to study the wave equation in general. We first show that it arises from the Dirichlet energy if we flip a sign… that may be important later! It is easy to solve the equation subject to initial data and periodic boundary data, since we may use Fourier series; we can also factor the wave equation, which gives the solution on $\mathbb R^n$ subject to initial data. We also prove finite propagation speed and the energy equipartition theorem, the latter a consequence of the Fourier transform. In addition, we show that while the Laplace equation $\Delta u = f$ satisfies the estimate $||u||_{L^2} \leq C||f||_{L^2}$, no such estimate is available for the wave equation, and I can say something vague here about the symbol being a nondegenerate conic section.

Now we recall the Michelson-Morley experiment, and deduce the causal properties of special relativity from the finite propagation speed of the wave equation. That funny Lagrangian really is the Dirichlet energy, and the wave equation really is the Laplace equation, provided that we get the relationship between space and time correct. As a corollary we deduce time dilation. Finally we introduce the Lorenz group and deduce that the Maxwell equation is invariant under it, thus showing that electricity and magnetism are one and the same.

Traffic flow and ideal fluids. We derive the Burgers equation as a simplified model for traffic flow on highways. We first observe that the Fourier transform doesn’t seem to actually help here, as the equation is quasilinear rather than linear.

We first attack the Burgers equation using the method of characteristics. We develop the theory of characteristics in general as a way to reduce a nonlinear first-order PDE to an ODE, before using them to solve the Burgers equation. We compute the blowup time, which gives a proof that subject to suitable initial data, traffic shockwaves must exist. We then give some numerical simulations to show the effect of traffic shockwaves on a traffic jam.

Having run into the issue of finite blowup time, we now go back to the Burgers equation and show that it is also the one-dimensional case of the Euler equation for an ideal fluid. Adding a viscosity term, we argue that the Burgers equation is actually a sort of nonlinear heat equation. Using the Cole-Hopf transform, we rewrite the Burgers equation as a heat equation and solve using the Fourier transform. We then talk about the vanishing-viscosity limit approximation to the traffic (inviscid) case, and when it is valid.

This lecture series would be somewhat open-ended, as students can come up with their own traffic flow models for the class to analyze, adding, e.g., terms to account for the possibility of an accident to Burgers’ equation.

Finite abelian groups and numerical analysis. Since I think an algebra class will not be a prerequisite, we first summarize the basics of finite abelian groups, and take their classification as a black box. We then discuss the Fourier transform on finite abelian groups, and argue that it converges to the Fourier series on a suitable torus in the large-order limit. Analogously to the theorem on $\mathbb R^n$-invariant PDE, we argue that circulant matrices are diagonalized by the Fourier transform. We then discuss the fast Fourier transform, and its numerical applications.

More on variational calculus. We develop the theory of variational calculus, and give sufficient conditions on a Lagrangian for its Euler-Lagrange equation to have a solution. This of course requires us to take the dominated convergence theorem and Sobolev trace theorems as black boxes. As applications, we show that many equations have solutions, and give a short proof of Brouwer’s fixed point theorem.

We then digress to define the notion of a Lie group. Since this is an undergraduate course, we just consider Lie groups which are smoothly embedded in $\mathbb R^n$. We then prove Noether’s theorem on the symmetries of Lagrangians, the “fundamental theorem of physics”.

Heisenberg’s uncertainty principle. We first introduce the Schrödinger equation and solve it using separation of variables and the Fourier transform. This reduces all questions of quantum mechanics to questions about the spectrum of the Hamiltonian, so we discuss quantum mechanics for a bit, and in particular how one can use the spectral theorem (taken as a black box) and the Laplacian to study quantum mechanics. This naturally leads to the question of “hearing the shape of a drum”, which is hard, so we do not try to answer it. If I remember correctly, however, one can at least prove Weyl’s law on the distribution of eigenvalues using fairly low-tech methods. We then prove Heisenberg’s uncertainty principle for functions in $L^2$. We do not, however, dare try to give a physical interpretation of it, other than the music-theoretic interpretation.