$L^p$-$L^{q}$-$L^{r}$ estimates and minimal decay regularity for   compressible Euler-Maxwell equations
  Due to the dissipative structure of \textit{regularity-loss}, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions to dissipative systems. The aim of this paper is to seek the lowest regularity index for the optimal decay rate of $L^{1}(\mathbb{R}^n)$-$L^2(\mathbb{R}^n)$. Consequently, a notion of minimal decay regularity for dissipative systems of regularity-loss is firstly proposed. To do this, we develop a new time-decay estimate of $L^p(\mathbb{R}^n)$-$L^{q}(\mathbb{R}^n)$-$L^{r}(\mathbb{R}^n)$ type by using the low frequency and high-frequency analysis in Fourier spaces. As an application, for compressible Euler-Maxwell equations with the weaker dissipative mechanism, it is shown that the minimal decay regularity coincides with the critical regularity for global classical solutions. Moreover, the recent decay property for symmetric hyperbolic systems with non-symmetric dissipation is also extended to be the $L^p$-version. 
Jiang Xu, Naofumi Mori, Shuichi Kawashima
math.AP