Decay character theory for partially dissipative hyperbolic systems of balance laws

The partially dissipative systems that characterize many physical phenomena were first pointed out by Godunov (1961), then investigated by Friedrichs-Lax (1971) who introduced the convex entropy, and later by Shizuta-Kawashima (1984,1985) who initiated a simple sufficient criterion ensuring the global existence of smooth solutions and their large-time asymptotics. There has been remarkable progress in the past several decades, through various different attempts. However, the decay character theory for partially dissipative hyperbolic systems remains largely open, as the Fourier transform of Green's function is generally not explicit in multi-dimensions. In this paper, we propose a general $L^p$ energy method and provide the positive answer to the unsolved question left in [5]. Under the convex entropy and Shizuta-Kawashima conditions, we pinpoint the critical regularity for the global solutions to the partially dissipative systems in the hybrid Besov spaces with different exponents on conservative and dissipative components in low and high frequencies, from the viewpoint of spectral analysis and the $L^1$ time integrability of Lipschitz bound. Furthermore, a new effective quantity $Ψ(t,x)$ capturing the interaction of conservative and dissipative components is introduced, which enables us to establish the sharp characterization of large-time asymptotic behavior. Finally, we also show that, with faster orders of convergence, the conservative and dissipative components of solutions can be asymptotically approximated by new profiles generated from parabolic systems subjected to $Ψ_0$ (the initial effective quantity), in the spirit of Chapman-Enskog expansion.