Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

Claudia Garetto,Michael Ruzhansky,Christian Jäh

doi:10.1007/s00208-018-1672-1

Claudia Garetto, Michael Ruzhansky + Show 1 more

Open Access

https://doi.org/10.1007/s00208-018-1672-1

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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 2times 2 and 3times 3 systems, complemented by several examples.

Highlights

Michael Ruzhansky was supported in parts by EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151
We can take any n ≥ 1 and we can assume that m ≥ 2 since in the case m = 1 there are no multiplicities and much more is known. It is well-known that even if all the coefficients in A and B depend only on time, due to multiplicities, the best one can hope for is the well-posedness of the Cauchy problem (1) in suitable classes of Gevrey spaces
The main questions that we address in this paper are: (Q1) Under what structural conditions on the zero order part B(t, x, Dx ) is the Cauchy problem (1) well-posed in C∞ or, even better, in suitable scales of Sobolev spaces?

Summary

Well-posedness in anisotropic Sobolev spaces

This section is devoted to proving the well-posedness of the Cauchy problem (1). For the reader’s convenience we first give a detailed proof in the cases m = 2 and m = 3. This will inspire us in proving Theorem 1. We note that the case m = 2 has been studied in [27] and we will briefly review its derivartion. First we collect a few results about Fourier integral operators that we will need in the sequel

Auxiliary remarks

The general case

First step or Schur step

The triangularisation procedure