On hyperbolicity and Gevrey well-posedness. Part three: a class of weakly hyperbolic systems

TL;DR

This paper establishes the well-posedness of certain weakly hyperbolic nonlinear PDE systems in Gevrey spaces for indices above 1/2, using approximate symmetrizers and energy estimates.

Contribution

It introduces a method for proving Gevrey well-posedness for a broad class of weakly hyperbolic systems with non-diagonalizable principal symbols.

Findings

Well-posedness in Gevrey spaces for indices > 1/2

Construction of approximate symmetrizers for principal symbols

Analysis of the sharpness of Gevrey index bounds

Abstract

We consider a class of weakly hyperbolic systems of first-order, nonlinear PDEs. Weak hyperbolicity means here that the principal symbol of the system has a crossing of eigenvalues, and is not uniformly diagonalizable. We prove the well-posedness of the Cauchy problem in the Gevrey regularity for all Gevrey indices greater than $1/2$. The proof is based on the construction of a suitable approximate symmetrizer of the principal symbol and an energy estimate in Gevrey spaces. We discuss both the generality of the assumption on the structure of the principal symbol and the sharpness of the lower bound of the Gevrey index.