Negative Sobolev regularity and optimal decay for the fractional Navier--Stokes--Poisson system

Abstract

We establish global well-posedness and sharp higher-order decay estimates for the fractional compressible Navier--Stokes--Poisson system in $\R^{3}$ with fractional Lam\'e--Navier dissipation $\Lame^\alpha u$ with $\alpha\in[1/2,1]$, and Riesz coupling $\Lambda^{2\beta}\phi=\rho-\bar{\rho}$ with $\beta\in[0,1]$, where $\Lambda:=(-\Delta)^{1/2}$ and $\bar{\rho}>0$ denotes the constant background density. For small perturbations of a constant equilibrium in $H^{N\alpha}$ with $N\geq3$ and $N\alpha>5/2$, we prove the global existence and uniqueness of classical solutions. Furthermore, we propagate the negative Sobolev norms with uniform-in-time bounds and derive the higher-order decay estimates, provided that the initial perturbation additionally belongs to $\dot{H}^{-\vartheta}$ for some $\vartheta\in[0,3/2)$. Moreover, we establish the optimality of these decay rates within the prescribed initial data class for the velocity and the Riesz potential in their natural energy norms. The density decay exponents are also optimal whenever the available regularity permits the full Riesz-induced gain. The analysis relies on a nonlinear energy structure adapted to both fractional dissipation and Riesz interaction, together with Sobolev interpolation. The optimality of the decay is proved by constructing smooth initial data localized at low frequencies and proving matching upper and lower decay bounds for the resulting solutions. These solutions show that the low-frequency linear dynamics including the Riesz-induced density gain, persists under nonlinear evolution.