New structural conditions on decay property with regularity-loss for symmetric hyperbolic systems with non-symmetric relaxation

This paper is concerned with the weak dissipative structure for linear symmetric hyperbolic systems with relaxation. The authors of this paper had already analyzed the new dissipative structure called the regularity-loss type in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239-266]. Compared with the dissipative structure of the standard type in [T. Umeda, S. Kawashima and Y. Shizuta, On the devay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math. 1 (1984) 435-457; Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985) 249-275], the regularity-loss type possesses a weaker structure in the high-frequency region in the Fourier space. Furthermore, there are some physical models which have more complicated structure, which we discussed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure of two hyperbolic relaxation models with regularity loss, Kyoto J. Math. 57(2) (2017) 235-292]. Under this situation, we introduce new concepts and extend our previous results developed in [Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal. 205 (2012) 239-266] to cover those complicated models.