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Hyperbolic systems with variable coefficients

We want to extend our previous analysis to linear systems of the form
This is the motivation for the definition of hyperbolicity (1.1.18) in the context of constant coefficient problems: freeze the coefficients and assume the hyperbolicity of the constant coefficient problem(s), tex2html_wrap_inline11059, uniformly for each tex2html_wrap_inline11061; then - in contrast to the notion of weak hyperbolicity, the variable coefficients problem is also hyperbolic. This result is based on the invariance of the notion of hyperbolicity under low-order perturbationsgif.

As before the study of the variable coefficients problem can be carried out by one of two ways:

: The symmetric hyperbolic case tex2html_wrap_inline11067: we can rewrite such symmetric problems in the equivalent form
In this case the symmetry of the tex2html_wrap_inline11069's implies that tex2html_wrap_inline11071 is skew-adjoint, i.e., integration by parts gives
Therefore we have
and hence the semi-boundedness requirement (hyper.26) holds with tex2html_wrap_inline11073. Consequently, if tex2html_wrap_inline11075 are symmetric (or at least symmetrizable) then the system (1.1.17) is hyperbolic.

Eitan Tadmor
Thu Jan 22 19:07:34 PST 1998