Abstract
Ducomet et al. (Discrete Contin Dyn Syst 11(1): 113–130, 2004) showed the existence of global weak solutions to the Navier–Stokes–Poisson system. We study the global behavior of such a solution. This is done by (1) proving uniqueness of a solution to the stationary system; (2) by showing convergence of a weak solution to the stationary solution. In (1) we consider only the case with repulsion. We prove our result in the case of a bounded domain with smooth boundary in \({\mathbb{R}^3}\) and also in the case of the whole space \({\mathbb{R}^3}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
DiPerna R.J., Lions P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Donatelli D.: Local and global existence for the coupled Navier-Stokes-Poisson problem. Quart. Appl. Math. 61(2), 345–361 (2003)
Ducomet B., Zlotnik A.: Stabilization and stability for the spherically symmetric Navier-Stokes-Poisson system. Appl. Math. Lett. 18(10), 1190–1198 (2005)
Ducomet B., Feireisl E., Petzeltová H., Straškraba I.: Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete Contin. Dyn. Syst. 11(1), 113–130 (2004)
Feireisl E., Petzeltová H.: Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow. Arch. Ration. Mech. Anal. 150(1), 77–96 (1999)
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Linearized steady problems, vol. I. In: Springer Tracts in Natural Philosophy, vol. 38. Springer, New York (1994)
Jiang F., Tan Z., Yan Q.: Asymptotic compactness of global trajectories generated by the Navier-Stokes-Poisson equations of a compressible fluid. Nonlinear Differ. Equ. Appl. 16(3), 355–380 (2009)
Kobayashi T., Suzuki T.: Weak solutions to the Navier-Stokes-Poisson equation. Adv. Math. Sci. Appl. 18(1), 141–168 (2008)
Li, H.-L., Matsumura, A., Zhang, G.: Optimal decay rate of the compressible Navier-Stokes-Poisson system in \({\mathbb{R}^3}\) (2008)
Matušů-Nečasová Š., Okada M., Makino T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. III. Jpn. J. Indust. Appl. Math. 14(2), 199–213 (1997)
Tan Z., Wang Y.: Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force. Nonlinear Anal. 71(11), 5866–5884 (2009)
Zhang Y., Tan Z.: On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 30(3), 305–329 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by I. Straskraba
Rights and permissions
About this article
Cite this article
Bella, P. Long Time Behavior of Weak Solutions to Navier–Stokes–Poisson System. J. Math. Fluid Mech. 14, 279–294 (2012). https://doi.org/10.1007/s00021-011-0051-4
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00021-011-0051-4