Energy estimates for the WR class

TL;DR

This paper develops robust energy estimates for weakly regular hyperbolic boundary value problems, addressing challenges posed by loss of regularity and paving the way for analyzing nonlinear problems.

Contribution

It introduces a more robust approach to derive a priori estimates for WR class problems, extendable to variable coefficients and nonlinear analysis.

Findings

Derived a priori estimates comparable to existing results

Developed a method extendable to scalar and system problems

Facilitated analysis of nonlinear problems using iterative schemes

Abstract

In this paper, we discuss energy estimates for a particular class of linear hyperbolic boundary value problems known as weakly regular of real type. Such class, also called WR in the literature, is relevant in many physical situations like the formation of shock waves in isentropic gas dynamics. In this and other WR examples, the failure of the uniform Lopatinskii condition plays a major role since it is associated with a loss of regularity in the scale of Sobolev spaces, eventually leading to energy inequalities that are ill-suited for nonlinear problems when solved by iteration. In the course of this work, we derive a priori estimates for the model case that are comparable to existing results, but using a more robust approach that we can extend to scalar or system problems with variable coefficients. The result is a significant step towards the ultimate goal of having estimates that can be combined with Piccard's or Newton's iterative scheme to analyze nonlinear situations.