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

TL;DR

This work analyzes the CNSP system in $\mathbb{R}^{d}$, $d\ge2$, with repulsive electrostatic coupling and reveals a Klein–Gordon–type dispersive effect at low frequencies. The authors develop new $L^{p}$-type linear estimates via a Green matrix that behave like heat diffusion with dispersion, enabling global well-posedness in the critical Besov space for $1\le p<2d$ without hyperbolic symmetrization or $L^{2}$-low-frequency data. They introduce a novel $L^{p}$-type low-frequency boundedness condition and a time-weighted Lyapunov framework to obtain optimal decay toward equilibrium, with density decaying faster than velocity due to the Poisson coupling. The results significantly broaden the admissible data and provide refined long-time behavior in a critical, partially dissipative setting, with potential applicability to other partially dissipative PDE models.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Proposition 2.1
• Proposition 2.2
• Lemma 2.1
• proof
• Proposition 3.1
• proof