Blog - fluid flows and infinite dimensional manifolds (part 5)

This page is a blog article in progress, written by Tim van Beek. To see discussions of this article while it was being written, visit the Azimuth Forum.

What is the simplest possible nonlinear partial differential equation? First, we will need derivatives of two different variables in the equation. If we have none, it won’t be a differential equation. And if we have only one, it will be an ordinary differential equation. So that gets us to write down

$u_t + u_x = 0$

We are using the common abbreviations $u_t = \partial_t u$ and $u_x = \partial_x u$. Our unkown function $u$ is a real valued function of two variables:

$u: \mathbb{R}^2 \to \mathbb{R}$

To make it nonlinear, we need to multiply on of the two summands by a function $f(u, u_x, u_t)$. To me, the simplest choice for $f$ is $f(u, u_x, u_t) = u$, so that gets us

$u_t + u \; u_x = 0$

This equation is called the Hopf or inviscid Burgers’ equation. I have mentioned it before: It is the geodesic equation of the group of diffeomorphisms of the circle, see the first part of the series.

I will interpret $t$ as time and $x$ as the spatial coordinate in the one dimensional space $\mathbb{R}$.

In this blog post I will talk a phenomen that is inherently nonlinear using this equation, which is the formation of shocks (TODO: maybe a reference here)? One interesting aspect of Burgers equation is that it is possible to obtain solutions in closed forms, because one can transform it into a *linear* equation. Do you see how? It took mathematicians some time to figure that out, so I assume it is not easy to see. I will tell you about it in this post, too.

For the moment let us consider the more general equation

$u_t + f(u) \; u_x = 0$

for some function $f(u)$. If $f$ is constant, that is $f(u) = c$, we get a linear equation:

$u_t + c \; u_x = 0$

This is not very interesting: If we start with a function $u_0(x)$ as an initial condition for $u$ at the time $t = 0$, then the solution of this equation consists simply of the translation of $u_0$ with velocity $c$ along the $x$ axis:

$u(x, t) = u_0(x - c t)$

If we interpret $u$ as describing some physical property of a wave that is evolving in time, this means that all features of the wave travel with the same velocity. If $f' \lt 0$, then higher values of $u$ will travel slower than smaller values. If $f' \gt 0$, then higher values will travel faster than smaller values. This is the case for the Burgers equation, so if we start with a bell curve as the initial profile $u_0(x)$, the solution to the Burgers equation will form a “shock” in finite time:

This picture is from

• G.B. Whitham: *Linear and Nonlinear Waves (Pure and Applied Mathematics*, John Wiley and Sons, New York, 1974

Note that in the picture $\rho$ is what I keep calling $u$. Mathematically, the solution $u$ becomes multivalued at a certain time $t_B$. While this may look very similar to a breaking wave on a beach,

… here is a word of caution: The solutions to the Burgers equation do not describe water waves. If the solution describes a physical property of a system, the formation of shocks actually tells us that the mathematical model breaks down. This means that some assumptions of the model become invalid. If we model a gas, for example, there will be effects from, e.g., thermodynamics that kick in and prevent the physical system from forming shocks.