Decay structure of two hyperbolic relaxation models with regularity-loss

TL;DR

This paper investigates decay structures of two hyperbolic relaxation models with regularity-loss, extending previous results by constructing more complex models and analyzing their energy dissipation properties using Fourier methods.

Contribution

It introduces two new complex models of regularity-loss type hyperbolic systems with detailed energy dissipation analysis and explicit construction of compensating matrices.

Findings

Established decay estimates for the new models.

Constructed explicit symmetric and skew-symmetric matrices for energy analysis.

Analyzed the dissipative structure in higher dimensions.

Abstract

The paper aims at investigating two types of decay structure for linear symmetric hyperbolic systems with non-symmetric relaxation. Precisely, the system is of the type $(p,q)$ if the real part of all eigenvalues admits an upper bound $-c|\xi|^{2p}/(1+|\xi|^2)^{q}$, where $c$ is a generic positive constant and $\xi$ is the frequency variable, and the system enjoys the regularity-loss property if $p<q$. It is well known that the standard type $(1,1)$ can be assured by the classical Kawashima-Shizuta condition. A new structural condition was introduced in \cite{UDK} to analyze the regularity-loss type $(1,2)$ system with non-symmetric relaxation. In the paper, we construct two more complex models of the regularity-loss type corresponding to $p=m-3$, $q=m-2$ and $p=(3m-10)/2$, $q=2(m-3)$, respectively, where $m$ denotes phase dimensions. The proof is based on the delicate Fourier energy method as well as the suitable linear combination of series of energy inequalities. Due to arbitrary higher dimensions, it is not obvious to capture the energy dissipation rate with respect to the degenerate components. Thus, for each model, the analysis always starts from the case of low phase dimensions in order to understand the basic dissipative structure in the general case, and in the mean time, we also give the explicit construction of the compensating symmetric matrix $K$ and skew-symmetric matrix $S$.