Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

TL;DR

This paper investigates the existence and uniqueness of generalized solutions to symmetric hyperbolic PDE systems with Colombeau coefficients, using refined energy estimates and extending previous results in the field.

Contribution

It introduces new solvability results for hyperbolic systems with generalized function coefficients, expanding the theoretical framework and connecting to recent research by Garetto and Oberguggenberger.

Findings

Established existence and uniqueness of solutions in Colombeau algebras.

Developed refined energy estimates for hyperbolic systems.

Extended previous results with new variants and partial cases.

Abstract

We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as coefficients and data. The proofs of solvability are based on refined energy estimates on lens-shaped regions with spacelike boundaries. We obtain several variants and also partial extensions of previous results and provide aspects accompanying related recent work by C. Garetto and M. Oberguggenberger.