Overview

For Burgers equation we used an iteration solving linear equations to obtain a local existence result. This technique carries over to first-order hyperbolic, to second-order parabolic and to mixed systems. The important point is again to obtain a priori estimates of the solution and its derivatives; these lead in turn to uniform estimates for the iteration sequence mentioned above.

Not to obscure the underlying principles, we shall not consider the most general situation, however, but restrict ourselves mainly to equations

where A= A(u) is a given smooth matrix function. As before, all (vector- and matrix-) functions are assumed to be real, for simplicity, and 1-periodic in x.

In Section 5.1 we treat the case where A(u) and all its derivatives are globally bounded. For ?>0, one obtains existence for all time. If the system

is hyperbolic, one obtains short time existence for ?=0.

The assumption of global boundedness of A(u) is, in general, not fulfilled in applications. In Section 5.2 we use a simple cut-off technique to treat more general cases and obtain short time existence results if A(u) is smooth in a neighborhood of the initial data.

If a finite time T ₀>0 is given ( T ₀ is not necessarily small), how can one decide whether a solution of (5.1.1) exists in 0 ? t ?T ₀? As for ordinary differential equations, this problem is often of a quantitative nature. We will sketch in...