Sharp decay characterization 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 provide a positive answer to the open question by means of the general $L^p$ energy method. Precisely, a new {\emph{effective quantity}} $\Psi(t,x)$ motivated by the compressible Euler system with damping is introduced, which enables us to capture leading diffusion profiles of the large-time behavior in the spirit of the Chapman-Enskog expansion. Consequently, we prove that the solutions approach the constant equilibrium state in the $\dot{\!B}^{\sigma}_{p,1}$-norm at the rate $t^{-(\sigma-\sigma_1)/2}$ as $t\rightarrow\infty$, and the corresponding norm of dissipative components decays at the enhanced rate $t^{-(\sigma-\sigma_1+1)/2}$, where the boundedness assumption in the $\dot{B}^{\sigma_1}_{p,\infty} (-d/p\leq \sigma_1<d/p-1$)-norm of the low frequencies of conservative components is not only sufficient, but also necessary to achieve those upper bounds of decay estimates. Furthermore, both upper and lower bounds for time-decay estimates are obtained if and only if the low-frequency part of $\Psi_0(x)$ (the initial effective quantity) is bounded in a non-trivial subset of $\dot{B}^{\sigma_1}_{p,\infty}$.