The optimal decay estimates on the framework of Besov spaces for generally dissipative systems

We give a new decay framework for general dissipative hyperbolic system and hyperbolic-parabolic composite system, which allow us to pay less attention on the traditional spectral analysis in comparison with previous efforts. New ingredients lie in the high-frequency and low-frequency decomposition of a pseudo-differential operator and an interpolation inequality related to homogeneous Besov spaces of negative order. Furthermore, we develop the Littlewood-Paley pointwise energy estimates and new time-weighted energy functionals to establish the optimal decay estimates on the framework of spatially critical Besov spaces for degenerately dissipative hyperbolic system of balance laws. Based on the $L^p({\mathbb{R}}^n)$ embedding and improved Gagliardo-Nirenberg inequality, the optimal $L^p({\mathbb{R}}^n)$-$L^2({\mathbb{R}}^n)(1\le p<2)$ decay rates and $L^p({\mathbb{R}}^n)$-$L^q({\mathbb{R}}^n)(1\le p<2\le q\le \mathrm{\infty })$ decay rates are further shown. Finally, as a direct application, the optimal decay rates for 3D damped compressible Euler equations are also obtained.