THE STABILITY OF STATIONARY SOLUTION FOR OUTFLOW PROBLEM ON THE NAVIER-STOKES-POISSON SYSTEM

doi:10.1016/S0252-9602(16)30058-3

Abstract

Abstract:

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.

Key words: Navier-Stokes-Poisson system, stationary solution, outflow problem, convergence rate, weighted energy method

CLC Number:

35B35

Mina JIANG, Suhua LAI, Haiyan YIN, Changjiang ZHU. THE STABILITY OF STATIONARY SOLUTION FOR OUTFLOW PROBLEM ON THE NAVIER-STOKES-POISSON SYSTEM[J].Acta mathematica scientia,Series B, 2016, 36(4): 1098-1116.

Trendmd

References

[1] Cui H B, Gao Z S, Yin H Y, et al. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line:existence, stability and convergence rate (submitted)
[2] Degond P. Mathematical modelling of microelectronics semiconductordevices. Some Current Topics on Nonlinear Conservation Laws, 2000, 15:77-110
[3] Donatelli D, Marcati P. A quasineutral type limit for the Navier-Stokes-Poisson system with large data. Nonlinearity, 2008, 21:135-148
[4] Duan R J, Ma H F. Global existence and convergence rates for the 3-D compressible Navier-Stokes equations without heat conductivity. Indiana Univ Math J, 2008, 5:2299-2319
[5] Duan R J, Yang X F. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Comm Pure Appl Anal, 2013, 12:985-1014
[6] Hao C C, Li H L. Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J Differential Equations, 2009, 246:4791-4812
[7] Hsiao L, Li H L. Compressible Navier-Stokes-Poisson equations. Acta Math Sci Ser B Engl Ed, 2010, 30:1937-1948
[8] Kawashima S, Nakamura T, Nishibata S, et al. Stationary waves to viscous heat-conductive gases in halfspace:existence, stability and convergence rate. Math Models Methods Appl Sci, 2010, 20:2201-2235
[9] Kawashima S, Nishibata S. Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries. Comm Math Phys, 2000, 211:183-206
[10] Kawashima S, Nishibata S, Zhu P C. Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space. Commun Math Phys, 2003, 240:483-500
[11] Kawashima S, Zhu P C. Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space. J Differential Equations, 2008, 244:3151-3179
[12] Lawrence P. Differential Equations and Dynamical Systems. Springer-Verlag, New York Inc, 2012
[13] Li H L, Matsumura A, Zhang G J. Optimal decay rate of the compressible Navier-Stokes-Possion system in R³. Arch Ration Mech Anal, 2010, 196:681-713
[14] Li H L, Yang T, Zou C. Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system. Acta Mathematica Scientia, 2009, 29B:1721-1736
[15] Liu Q Q, Yin H Y, Zhu C J. Stability of contact discontinuity for the Navier-Stokes-Poisson system with free boundary (submitted)
[16] Matsumura A. Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas. Methods Appl Anal, 2001, 8:645-666
[17] Matsumura A, Nishihara K. Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. Comm Math Phys, 1994, 165:83-96
[18] Nakamura T, Nishibata S, Yuge T. Convergence rate of solutions toward stationary solution to the compressible Navier-Stokes equation in a half line. J Differential Equations, 2007, 241:94-111
[19] Nikkuni Y, Kawashima S. Stability of stationary solutions to the half space problem for the discrete Boltzmann equation with multiple collisions. Kyushu J Math, 2000, 54:233-255
[20] Nishikawa M. Convergence rate to the traveling wave for viscous conservation laws. Funkcial Ekvac, 1998, 41:107-132
[21] Ruan L Z, Yin H Y, Zhu C J. The stability of the superposition of rarefaction wave and contact discontinuity for the Navier-Stokes-Poisson system with free boundary (submitted)
[22] Wang S, Jiang S. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Comm Partial Differential Equations, 2006, 31:571-591
[23] Wang W K, Wu Z G. Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multidimensions. J Differential Equations, 2010, 248:1617-1636
[24] Wu Z G, Wang W K. Pointwise estimates of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions. Acta Mathematica Scientia, 2012, 32B:1681-1702
[25] Yin H Y, Zhang J S, Zhu C J. Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system. Nonl Anal:Real World Appl, 2016, 31:492-512
[26] Zhang G J, Li H L, Zhu C J. Optimal decay rate of the non-isentropic compressible Navier-Stokes-Possion system in R³. J Differential Equations, 2011, 250:866-891
[27] Zhou F, Li Y P. Convergence rate of solutions toward stationary solutions to the bipolar Navier-StokesPoisson equations in a half line. Bound Value Probl, 2013, 124:1-22