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# Communications in Mathematical Sciences

## Volume 17 (2019)

### Number 3

### Generalization of Kreiss theory to hyperbolic problems with boundary-type eigenmodes

Pages: 669 – 703

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n3.a5

#### Author

#### Abstract

The Kreiss symmetrizer technique gives sharp estimates of the solutions of the firstorder hyperbolic initial-boundary value problems both in the interior and at the boundary of the domain. Such estimates imply robustness and strong well-posedness in the generalized sense, and the corresponding problems are called strongly boundary stable, satisfying the Kreiss eigenvalue condition. There are however problems that are not strongly boundary stable and yet are well-posed and robust. For such problems sharp estimates of the solution can be obtained only in the interior and not at the boundary. We refer to this class of problems as well-posed in the generalized sense. Examples include hyperbolic problems, governed by elastic and Maxwell’s equations, that describe boundary-type wave phenomena, such as surface waves and glancing waves. We introduce the notion of boundary-type generalized eigenvalues and obtain a sufficient algebraic condition for well-posedness in the generalized sense, thereby relaxing the Kreiss eigenvalue condition. Despite the utilization of the Laplace–Fourier mode analysis, since the proofs are based on the construction of smooth Kreiss-type symmetrizers, the developed theory can be applied to problems with variables coefficients in both first-order and second-order forms.

#### Keywords

Kreiss theory, hyperbolic systems, well-posedness, Kreiss symmetrizers, boundary stability, boundary phenomena, surface waves, glancing waves

#### 2010 Mathematics Subject Classification

35A02, 35C07, 35L50, 35Q60, 35Q86

Dedicated to the memory of Heinz-Otto Kreiss.

Received 17 March 2018

Accepted 22 January 2019

Published 30 August 2019