The results of Chapter 3 about linear parabolic, strongly hyperbolic, and mixed systems can mostly be generalized from one space dimension to an arbitrary number of space dimensions. These generalizations are carried out in the first three sections of this chapter. In particular, we show well-posedness of the Cauchy problem for the linearized compressible Navier-Stokes and Euler equations. Section 6.4 treats short-time existence for nonlinear systems; we carry out the details for the symmetric hyperbolic case and sketch generalizations to systems of different type. For a special class of nonlinear parabolic systems in two space dimensions, we will prove all-time existence in Section 6.5.

6.1. Linear Parabolic Systems

We consider second-order parabolic systems of the general form

Here s is the number of space dimensions and D _i is the operator D _i= ?/ ? x _i . The coefficient matrices A _ij= A _ij (x, t), B _i =B _i (x, t), C=C(x, t) and the forcing function F=F(x, t) are assumed to be C ^?-smooth, real, and 1-periodic in each component x _i, i=1, , s. The concept of parabolicity will be defined below. For the function u(x, t) an initial condition

is given, where f takes values in R ⁿ. As before, we seek a smooth solution u= u(x, t) taking values in R ⁿ which is 1-periodic in each x _i , i