Introduction

We now focus on hyperbolic theory. It the present chapter 6 we look at a first order system of pseudodifferential equations on ℝn of ν equations in ν unknown functions, of the form

with a ν×ν-matrix of symbols k=((kj1)), kj1 ∈ ψce, for fixed t. In essence ‘hyperbolic’ means that either the matrix k(t,x,ξ) is self-adjoint, or at least has real eigenvalues, both mod ψ⊆0.

In seel we discuss a symmetric hyperbolic first order system of PDE's, using a method of mollifiers, after K.O.Friedrichs. The case of a ψdo K(t) may be treated by a similar technique (sec. 2). Here we use the weighted L2-Sobolev spaces of ch.III.

In sec.3 we will look at properties of the evolution operator U=U(τ,t). We find that U(τ,t) is of order 0, while ∂pt∂qτU is Of order (p+q)e. In sec.4 we discuss strictly hyperbolic systems, no longer symmetric hyperbolic, by the method of symmetrizer.

In sec.5 we discuss a global Egorov type result for a single hyperbolic equation, proving existence of a characteristic flow. Actually our flows are of more general type, called particle flow, using a generalized principal symbol of K, no longer homogeneous in ξ.

This flow is related to the family A <-> U−1 AU of automorphisms of the C* -subalgebra A of L (H) generated by Opψ⊆0 (V,lO), where U = U(τ,t) is the evolution operator: The restriction of the flow to |x| + |ξ|=∞ is the dual of the induced automorphism of Av = A/k(H). In sec.6 we discuss the action of these flows on symbols.