Weakly Hyperbolic Systems by Symmetrization

TL;DR

This paper introduces a novel symmetrization method for hyperbolic first order systems, providing a new approach to establish Gevrey well-posedness and clarifying spectral effects beyond eigenvalue multiplicities.

Contribution

It proposes a new symmetrization technique inspired by Lyapunov functions, offering a straightforward proof of Gevrey well-posedness for hyperbolic systems.

Findings

Provides a new symmetrizer construction

Simplifies a priori estimates for hyperbolic systems

Clarifies spectral effects on well-posedness

Abstract

We study hyperbolic first order systems and propose a new method proving Gevrey well posedness, constructing a symmetrizer, motivated by a special Lyapunov function for linear ODE. The proof not only gives a priori estimates straightforward so simply but also clarifies some effects coming from the spectral structures other than the multiplicities of the eigenvalues.