Overview

The Cauchy problem for the viscous and the inviscid Burgers equation will be discussed in detail in this chapter. The techniques we apply can mostly be generalized in a straightforward way to more complicated nonlinear parabolic or hyperbolic equations, to systems in one or more space dimensions. These generalizations will be carried out in Chapters 5 and 6. In other words, we will use Burgers equation as a simple example to illustrate a number of general techniques for the treatment of nonlinear evolutionary systems. Intentionally, we do not apply the celebrated Cole-Hopf transformation, which reduces Burgers equation to the heat equation. (Though the Cole-Hopf transformation does have interesting generalizations also, these seem to be too limited, at present, to discuss the Navier-Stokes system.) We emphasize in this chapter:

1. local (in time) existence of solutions via a linear iteration;

2. local existence together with global (in time) a priori estimates leads to global existence;

3. smoothing properties for the parabolic, i.e., viscous case;

4. breakdown of smooth solutions for the hyperbolic, i.e., inviscid case in finite time.

Concerning these points, our discussion of Burgers equation is representative for parabolic and hyperbolic systems.

With regard to other aspects, namely global a priori estimates, the maximum principle, and our discussion of shocks, the techniques for Burgers equation do not readily generalize. The difficulties involved will become apparent in Chapters 5 and 6.

4.1. Burgers Equation: A Priori Estimates and Local Existence

In this section we start our discussion of...