Dissipative structure of higher order regularizations of hyperbolic systems of conservation laws in several space dimensions
Felipe Angeles, Ramón G. Plaza, José Manuel Valdovinos
TL;DR
The paper develops a comprehensive framework to extend Kawashima–Shizuta type dissipativity to higher-order, multi-dimensional linear systems arising from regularizations of hyperbolic conservation laws. By adopting Humpherys' symbolic approach and introducing symbol symmetrizability, genuine coupling, and compensating matrix symbols, it proves an equivalence between strict dissipativity, genuine coupling, and compensating symbols, even without constant-multiplicity assumptions. The theory yields concrete pointwise Fourier-space energy estimates and decay rates, and it is applied to a suite of physical models (NSK, NSFK, EFK, DNSF, QHD), confirming dissipative structures in many but not all cases. Notably, multidimensional inviscid Korteweg-type and quantum-hydrodynamics systems can fail the hypotheses, delineating the framework’s limits. Overall, the work provides systematic tools for analyzing linearized dissipative structures of high-order, multi-dimensional regularizations with broad physical relevance.
Abstract
This work studies the dissipative structure of regularizations of any order of hyperbolic systems of conservation laws in several space dimensions. It is proved that the seminal equivalence theorem by Kawashima and Shizuta (Hokkaido Math. J. 14, 1985, no. 2, 249-275), which relates the strictly dissipative structure of second-order (viscous) systems to a genuine coupling condition of algebraic type, can be extended to higher-order multidimensional systems. For that purpose, the symbolic formulation of the genuine coupling condition by Humpherys (J. Hyperbolic Differ. Equ. 2, 2005, no. 4, 963-974) for linear operators of any order in one dimension, is adopted and extrapolated. Therefore, the concepts of symbol symmetrizability and genuine coupling are extended to the most general setting of differential operators of any order in several space dimensions. Applications to many viscous-dispersive systems of physical origin, such as compressible viscous-capillar fluids of Korteweg type, the dispersive Navier-Stokes-Fourier system and the equations of quantum hydrodynamics, illustrate the relevance of this extension.