Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy

TL;DR

This paper establishes the existence of smooth and weak solutions for the 3D compressible Navier-Stokes-Poisson equations with finite energy, and provides a criterion for solution blow-up based on density norms.

Contribution

It proves the existence of smooth solutions with small L^2-norm without smallness constraints on the H^4-norm, and introduces a blow-up criterion related to density.

Findings

Existence of smooth solutions with small L^2-norm and bounded densities.

Existence of weak solutions possibly with discontinuities.

Blow-up criterion based on the L-infinity norm of density.

Abstract

We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small $L^2$-norm and essentially bounded densities. No smallness assumption is imposed on the $H^4$-norm of the initial data. Using a compactness argument, we further obtain the existence of weak solutions which may have discontinuities across some hypersurfaces in $\mathbb{R}^3$. We also provide a blow-up criterion of solutions in terms of the $L^\infty$-norm of density.