Weak time-periodic solutions to the compressible Navier–Stokes–Poisson equations

Abstract

The compressible Navier-Stokes-Poisson equations driven by a time-periodic external force are considered in this paper. The system takes into account the effect of self-gravitation. We establish the existence of weak time-periodic solutions on condition that the adiabatic constant satisfies γ > 5 3 .

Global weak solutions to 3D compressible Navier–Stokes–Poisson equations with density-dependent viscosity

“…Inspired by [17], in the present paper, we consider the compressible Navier-Stokes equations with degenerate viscosities, potential force and the damping term, in which the pressure P (ρ) is not necessary a monotone function, and we prove that the problem admits a global weak solution as γ > 4 3 for the case λ = −1, or γ > 1 for the case λ = 1. Compared to the case of the classical compressible Navier-Stokes equations, the Navier-Stokes-Poisson problem is much more complex and some new difficulties will occur, for example, we can not deduce the energy estimates directly due to the poisson term and non-monotone pressure term, and moreover, when we deduce the energy estimates and the Bresch-Dejardins entropy, the estimates will depend on the index ε, δ and η, so we need to be very careful as we deduce these estimates because we need to tend the ε, η, δ to zero step by step later in the proof of the main theorem.…”

“…Zhang and Tan in [24], by using the theory of Orlicz spaces, have proved the existence of globally defined finite energy weak solutions for Navier-Stokes-Poisson equations in two dimensions with the pressure satisfying P (ρ) = aρ log d ρ for large ρ, and d > 1, a > 0. Cai and Tan [4] also proved the system has the global weak time-periodic solution for the Navier-Stokes-Poisson equations in a bounded domain with periodic boundary condition as γ > 5 3 when the external force is time-periodic. Besides, Jiang et al [16] considered the global behavior of weak solutions of the Navie-Stokes-Poisson equations in a bounded domain with arbitrary forces.…”

“…However, for the case of density-dependent viscosity coefficients, the problem is much more challenge because of the degeneration near the vacuum and the results are limit. Ducomet et.al in [8] studied the global stability of the weak solutions to the Navier-Stokes-Poisson equations with the degenerate viscosities as γ > 4 3 , in which the pressure P (ρ) is not necessary a monotone function of the density. Furthermore, in [9] they also considered Cauchy problem for the Navier-Stokes-Poisson equations of spherically symmetric motions in R 3 , both constant viscosities and density-dependent viscosities included, and proved the global stability of the weak solutions just provided that the polytropic index γ satisfies γ > 1.…”