Global existence of strong solutions of Navier-Stokes-Poisson equations for one-dimensional isentropic compressible fluids

Abstract

The authors prove two global existence results of strong solutions of the isen-tropic compressible Navier-Stokes-Poisson equations in one-dimensional bounded intervals. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition. In this paper the initial vacuum is allowed, and T is bounded.

Keywords

Global strong solutions / Navier-Stokes-Poisson equations / Existence and uniqueness

References

[1] Kobayashi, T. and Suzuki, T., Weak solutions to the Navier-Stokes-Poisson equation, preprint.

[2] Feireisl E., Novotný A., Petzeltová H.. On the existence of globally defined weak solution to the Navier-Stokes equations. J. Math. Fluid Mech., 2001, 3: 358-392

[3] da Veiga Beirão H.. Long time behavior for one-dimensional motion of a general barotropic viscous fluid. Arch. Rat. Mech. Anal., 1989, 108: 141-160

[4] Kazhikhov A. V.. Stabilization of solutions of an initial-boundary-value problem for the equations of motion of a barotropic viscous fluid. Diff. Eqs., 1979, 15: 463-467

[5] Solonnikov V. A., Kazhikhov A. V.. Existence theorem for the equations of motion of a compressible viscous fluid. Ann. Rev. Fluid Mech., 1981, 13: 79-95

[6] Straškraka I., Valli A.. Asymptotic behavior of the density for one-dimensional Navier-Stokes equations. Manuscripta Math., 1988, 62: 401-416

[7] Matsumura A., Yanagi S.. Uniform boundedness of the solutions for a one dimensional isentropic model system of compressible viscous gas. Comm. Math. Phys., 1996, 175: 259-274

[8] Jiang S., Zhang P.. On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Comm. Math. Phys., 2001, 215: 559-581

[9] Choe H. J., Kim H.. Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Diff. Eqs, 2003, 190: 504-523

[10] Choe H. J., Kim H.. Global existence of the radially symmetric solutions of the radially symmetric solutions of the Navier-Stokes equation for the isentropic compressible fluids. Math. Meth. Appl. Sci., 2005, 28: 1-28

[11] Lions P. L.. Mathematical topics in fluids mechanics. Oxford Lecture Series in Mathematics and Its Applications, 1998, Oxford: Clarendon Press

[12] Xin Z.. Blow up of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math., 1998, 51: 229-240

[13] Hoff D.. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Diff. Eqs., 1995, 120: 215-254

[14] Hoff D., Smoller J.. Non-formation of vacuum states for compressible Navier-Stokes equations. Comm. Math. Phys., 2001, 216: 255-276

[15] Liu T. P., Xin Z., Yang T.. Vacuum states of compressible flow. Discrete and Continuous Dyn. Systems, 1998, 4: 1-32

[16] Luo T., Xin Z., Yang T.. Interface behavior of compressible Navier-Stokes equations with vacuum. SIAM J. Math. Anal., 2000, 31(6): 1175-1191

[17] Yang T., Yao Z., Zhu C.. Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Comm. PDE, 2001, 26(5-6): 965-981

[18] Okada M., Makino T.. Free boundary problem for the equation of spherically symmetric motion of viscous gas. Japan J. Indust. Appl. Math., 1993, 10: 219-235

[19] Matsusu-Necasova S., Okada M., Makino T.. Free boundary problem for the equation of spherically symmetric motion of viscous gas. (The Third) Japan J. Indust. Appl. Math., 1997, 14: 199-213