Electrostatic effects on critical regularity and long-time behavior of viscous compressible fluids

TL;DR

This paper proves the global well-posedness and analyzes the long-time behavior of solutions to the compressible Navier-Stokes-Poisson equations, highlighting the impact of electrostatic coupling on fluid dynamics and decay rates.

Contribution

It introduces a novel analysis that removes previous frequency restrictions, allowing highly oscillatory initial velocities, and establishes optimal decay rates for solutions with broader initial data.

Findings

Global well-posedness in critical Besov spaces without hyperbolic symmetrization.

Extension of initial data range to include highly oscillatory velocities.

Establishment of faster decay rates under generalized low-frequency assumptions.

Abstract

We consider the compressible Navier-Stokes-Poisson equations in $\mathbb{R}^d$ ($d\geq2$), a classical model for barotropic compressible flows coupled with a self-consistent electrostatic potential. We show that the electrostatic coupling has a significant impact on the long-time dynamics of solutions due to its underlying Klein-Gordon structure. As a first result, we prove the global well-posedness of the Cauchy problem with initial data near equilibrium in the full-frequency $L^{p}$-type critical Besov space \emph{without relying on hyperbolic symmetrization}. Compared with the Poisson-free case studied in several milestone works [Charve and Danchin, Arch. Rational Mech. Anal., 198 (2010), 233-271; Chen, Miao and Zhang, Commun. Pure Appl. Math., 63 (2010), 1173-1224; Haspot, Arch. Rational Mech. Anal., 202 (2011), 427-460], we remove the extra $L^{2}$ assumption in low frequencies and extend the admissible choice of $p$ to the sharp range $1\leq p<2d$. This is, to the best of our knowledge, the first result in compressible fluids that allows the initial velocity field to be highly oscillatory across all frequencies. Furthermore, stemming from the Poisson coupling, the density and velocity exhibit distinct low-frequency behaviors. Motivated by this feature, we propose a general $L^p$-type low-frequency assumption and establish the optimal convergence rates of global solutions toward equilibrium. For a broad class of indices, this assumption yields faster decay than those obtained under the classical $L^1$ framework. To this end, we develop a time-weighted energy method, which is of interest and enables us to capture maximal decay estimates without additional smallness of initial data.