Global classical solutions to partially dissipative hyperbolic systems violating the Kawashima condition

TL;DR

This paper establishes the existence of global smooth solutions for certain partially dissipative hyperbolic systems in multiple dimensions, even when the Kawashima condition is violated, by imposing additional degeneracy conditions.

Contribution

It introduces a novel framework with partially normalized coordinates and refined energy estimates to handle systems violating the Kawashima condition.

Findings

Global classical solutions are constructed near constant equilibria.

The method handles systems with eigen-family violating the Kawashima condition.

Decay estimates for dissipative components are refined.

Abstract

This paper considers the Cauchy problem for the quasilinear hyperbolic system of balance laws in $\mathbb{R}^d$, $d\ge 2$. The system is partially dissipative in the sense that there is an eigen-family violating the Kawashima condition. By imposing certain supplementary degeneracy conditions with respect to the non-dissipative eigen-family, global unique smooth solutions near constant equilibria are constructed. The proof is based on the introduction of the partially normalized coordinates, a delicate structural analysis, a family of scaled energy estimates with minimum fractional derivative counts and a refined decay estimates of the dissipative components of the solution.