In this chapter we treat second-order parabolic and first-order strongly hyperbolic systems in one space dimension. Instead of considering the Cauchy problem with initial data in L ₂, we deal with problems which are 1-periodic in x. The periodic problem has the technical advantage that the behavior at x= ? need not be specified, but the arguments for initial data in L ₂ would be essentially the same.

Assuming the existence of a smooth solution for the parabolic equation, we first prove estimates of the solution and its derivatives. In Section 3.2 we write down a simple difference scheme and prove analogous estimates for it. By sending the step-size to zero, we obtain a solution for parabolic systems in a rather elementary and constructive way. Strongly hyperbolic equations are treated by adding a small second-order term, whose coefficient is sent to zero. In a similar fashion as in the constant-coefficient case, we also treat certain mixed hyperbolic-parabolic systems and give an application to the linearized N-S equations.

In the parabolic case we use as a guiding principle: first, assume the existence of a solution and show estimates for it and its derivatives; second, write down a difference scheme and show analogous estimates which imply the existence of a solution. This principle is very useful for equations of different type, also. We demonstrate this in Section 3.6 with an application to the linearized Korteweg de Vries equation. The linear Schr dinger equation will...