Dissipative structure of higher-order regularizations of hyperbolic systems of conservation laws in several space dimensions

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 Shizuta and Kawashima [Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14(2) (1985) 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 multi-dimensional systems. For that purpose, the symbolic formulation of the genuine coupling condition by Humpherys [Admissibility of viscous-dispersive systems, J. Hyperbolic Differ. Equ. 2(4) (2005) 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 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.