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

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.

💡 Research Summary

The paper studies the compressible Navier‑Stokes‑Poisson (CNSP) system in $\mathbb{R}^{d}$ ($d\ge2$), a model that couples barotropic compressible fluid dynamics with a self‑consistent electrostatic potential. The authors focus on two fundamental questions: (i) global well‑posedness for small perturbations of a constant equilibrium in a critical functional framework, and (ii) optimal large‑time decay rates under a new low‑frequency hypothesis that exploits the Klein‑Gordon structure induced by the Poisson coupling.

Main contributions

1. Global existence in full‑frequency $L^{p}$‑critical Besov spaces.
   Theorem 1.1 proves that if the initial density perturbation $\rho_{0}-\rho^{*}$ belongs to $\dot B^{d/p-2}{p,1}\cap\dot B^{d/p}{p,1}$ and the initial velocity $u_{0}$ belongs to $\dot B^{d/p-1}_{p,1}$, with a small norm, then the Cauchy problem admits a unique global solution. The admissible range of the integrability exponent is the sharp interval $1\le p<2d$, which dramatically enlarges the previously known range $p\in