Smoothing Solutions to Initial-Boundary Problems for First-Order Hyperbolic Systems

TL;DR

This paper investigates the smoothing behavior of solutions to initial-boundary problems for linear first-order hyperbolic systems, establishing criteria to determine when solutions become infinitely differentiable over time.

Contribution

It introduces a general criterion for smoothing solutions in hyperbolic systems with complex boundary conditions, including cases with singular initial data.

Findings

Existence of unique delta wave solutions for strongly singular initial data

Criteria for solutions to become infinitely differentiable (smoothing)

Application to classical boundary conditions

Abstract

We consider initial-boundary problems for general linear first-order strictly hyperbolic systems with local or nonlocal nonlinear boundary conditions. While boundary data are supposed to be smooth, initial conditions can contain distributions of any order of singularity. It is known that such problems have a unique continuous solution if the initial data are continuous. In the case of strongly singular initial data we prove the existence of a (unique) delta wave solution. In both cases, we say that a solution is smoothing if it eventually becomes $k$-times continuously differentiable for each $k$. Our main result is a criterion allowing us to determine whether or not the solution is smoothing. In particular, we prove a rather general smoothingness result in the case of classical boundary conditions.