Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I. Well-posedness

TL;DR

This paper investigates the well-posedness of hyperbolic systems with non-diagonalisable principal parts and variable multiplicities, establishing conditions for triangularisation and solutions in anisotropic Sobolev spaces.

Contribution

It introduces new conditions for well-posedness of complex hyperbolic systems with variable multiplicities and Jordan block structures, including explicit triangularisation procedures.

Findings

Well-posedness in anisotropic Sobolev spaces for upper triangular systems.

Conditions for Schur triangularisation of systems with variable coefficients.

Explicit examples for 2x2 and 3x3 systems demonstrating the theory.

Abstract

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance $k$ to it must be of order $-k$. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for $2\times 2$ and $3\times 3$ systems, complemented by several examples.