Global existence and asymptotic decay for small solutions of general quasilinear hyperbolic balance laws
Matthias Sroczinski
TL;DR
The paper addresses global existence and decay for small solutions to general quasilinear hyperbolic balance laws without relying on an entropy structure or symmetry in the source term. It develops a para-differential framework under (H), (RH) and the dissipation conditions (D1)-(D3) to capture dissipation across Fourier scales, establishing linear Fourier-space decay and a nonlinear stability result in $H^s\cap L^1$ for $d\ge3$ with rate $(1+t)^{-d/4}$. The results apply to multidimensional Jin–Xin relaxation systems and clarify the relationship to entropy-dissipative theories, showing global stability beyond entropy-based methods. This extends classical Kawashima–Yong-type results to a broader class of systems and provides a robust tool for analyzing nonlinearly dissipative hyperbolic balance laws in higher dimensions.
Abstract
This paper establishes global existence and asymptotic decay for small solutions to quasilinear systems of hyperbolic balance laws, where, generalizing previous works, the hyperbolic operator does not need to admit an entropy nor does the source term need to satisfy any symmetry assumptions. Dissipative properties are characterized by three conditions corresponding to regimes of small, intermediate and large wave numbers in Fourier space and the fully non-linear system is treated by using methods of para-differential calculus recently developed in the context for proofs of global existence and decay in second-order hyperbolic systems. The present work leads, in particular, to asymptotic stability of rest-states for multidimensional Jin-Xin relaxation system, a result not accessible through previous methods.