Nonlinear hyperbolic systems

In this chapter we treat various types of hyperbolic equations, beginning in §5.1 with first order symmetric hyperbolic systems. In this case, little direct use of pseudodifferential operator techniques is made, mainly an appeal to the Kato-Ponce estimates. We use Friedrichs mollifiers to set up a modified Galerkin method for producing solutions, and some of their properties, such as (5.1.43), can be approached from a pseudodifferential operator perspective. The idea to use Moser type estimates and to aim for results on persistence of solutions as long as the C1-norms remain bounded was influenced by [Mj]. We provide a slight sharpening, demonstrating persistence of solutions as long as the C1_*-norm is bounded. In §5.2 we study two types of symmetrizable systems, the latter type involving pseudodifferential operators in an essential way. Here and in subsequent sections, including a treatment of higher order hyperbolic equations, we make strong use of the C1S_clm-calculus developed in Chapter 4.