Fredholmness and Smooth Dependence for Linear Time-Periodic Hyperbolic System

TL;DR

This paper establishes conditions for Fredholm solvability and smooth dependence of solutions on data for linear time-periodic hyperbolic systems, including cases with discontinuous coefficients and non-strict hyperbolicity.

Contribution

It provides new criteria ensuring Fredholmness, unique solvability, and smooth data-to-solution maps for hyperbolic systems with various coefficient regularities.

Findings

Fredholm solvability conditions derived for hyperbolic systems.

Existence and uniqueness of solutions under specific data conditions.

Analysis of how coefficient perturbations affect system properties.

Abstract

This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type $$ \partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,...,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data $a_j$ and $b_{jk}$ and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution-map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients $a_j$ lead to essentially different results than perturbations of the coefficients $b_{jk}$, in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients $a_j$ and $b_{jk}$, but they are new even in the case of strict hyperbolicity and of smooth coefficients.