On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in $\Bbb R^N$

TL;DR

This paper proves that under certain integrability and asymptotic conditions, there are no global weak solutions to the stationary and time-dependent Euler-Poisson and Navier-Stokes-Poisson equations in multi-dimensional space.

Contribution

It establishes nonexistence results for stationary and time-dependent weak solutions to these equations under broad assumptions, extending previous nonexistence results.

Findings

No stationary weak solutions under integrability conditions.

Nonexistence of solutions for time-dependent equations with asymptotic density behavior.

Results hold for dimensions N ≥ 2.

Abstract

In this paper we prove nonexistence of stationary weak solutions to the Euler-Poisson equations and the Navier-Stokes-Poisson equations in $\Bbb R^N$, $N\geq 2$, under suitable assumptions of integrability for the density, velocity and the potential of the force field. For the time dependent Euler-Poisson equations we prove nonexistence result assuming additionally temporal asymptotic behavior near infinity of the second moment of density. For a class of time dependent Navier-Stokes-Poisson equations this asymptotic behavior of the density can be proved if we assume the standard energy inequality, and therefore the nonexistence of global weak solution follows from more plausible assumption in this case.