Dissipative structure for symmetric hyperbolic-parabolic systems with Korteweg-type dispersion

Abstract

In this paper, we are concerned with generally symmetric hyperbolic-parabolic systems with Korteweg-type dispersion. Referring to those classical efforts by Kawashima et al., we formulate new structural conditions for the Korteweg-type dispersion and develop the dissipative mechanism of “regularity-gain type.” As an application, it is checked that several concrete model systems (e.g., the compressible Navier-Stokes(-Fourier)-Korteweg system) satisfy the general structural conditions. In addition, the optimality of our general theory on the dissipative structure is also verified by calculating the asymptotic expansions of eigenvalues.

Keywords:

• Decay property
• dissipative structure
• Euler-Fourier-Korteweg system
• hyperbolic-parabolic systems
• Korteweg-type dispersion
• Navier-Stokes-Fourier-Korteweg system

Mathematical Subject Classification 2010:

• 35G35
• 35Q35
• 35B35
• 35B40

Additional information

Funding

This work is partially supported by Top Global University Project and Toyota Central Research Institute Joint Research Fund. S. Kawashima is partially supported by JSPS KAKENHI Grant Numbers JP18H01131, JP19H05597, and JP20H00118. Y. Shibata is partially supported by JSPS KAKENHI Grant Number JP17H0109. J. Xu is partially supported by the National Natural Science Foundation of China (11871274, 12031006) and the China Scholarship Council (201906835023).