# American Institute of Mathematical Sciences

March  2013, 12(2): 985-1014. doi: 10.3934/cpaa.2013.12.985

## Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations

 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong 2 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received  October 2011 Revised  February 2012 Published  September 2012

In this paper, we are concerned with the initial boundary value problem on the two-fluid Navier-Stokes-Poisson system in the half-line $R_+$. We establish the global-in-time asymptotic stability of the rarefaction wave and the boundary layer both for the outflow problem under the smallness assumption on initial perturbation, where the strength of the rarefaction wave is not necessarily small while the strength of the boundary layer is additionally supposed to be small. Here, the large initial data with densities far from vacuum is also allowed in the case of the non-degenerate boundary layer. The results show that the large-time behavior of solutions coincides with the one for the single Navier-Stokes system in the absence of the electric field. The proof is based on the classical energy method. The main difficulty in the analysis comes from the slower time-decay rate of the system caused by the appearance of the electric field. To overcome it, we use the coupling property of the two-fluid equations to capture the dissipation of the electric field interacting with the nontrivial asymptotic profile.
Citation: Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
##### References:
 [1] C. Besse, J. Claudel and P. Degond, et al., A model hierarchy for ionospheric plasma modeling,, Math. Models Methods Appl. Sci., 14 (2004), 393.  doi: 10.1142/S0218202504003283.  Google Scholar [2] D. Chae, On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in $R^N$,, Comm. Partial Differential Equations, 35 (2010), 535.  doi: 10.1080/03605300903473418.  Google Scholar [3] S. Cordier, P. Degond, P. Markowich and C. Schmeiser, Travelling wave analysis of an isothermal Euler-Poisson model,, Ann. Fac. Sci. Toulouse Math., 5 (1996), 599.  doi: 10.1.1.57.5963.  Google Scholar [4] D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem,, Quart. Appl. Math., 61 (2003), 345.  doi: 10.1.1.14.6956.  Google Scholar [5] D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data,, Nonlinearity, 21 (2008), 135.  doi: 10.1088/0951-7715/21/1/008.  Google Scholar [6] R.-J. 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Anal., 197 (2010), 89.  doi: 10.1007/s00205-009-0267-0.  Google Scholar [11] F. Huang and X. Qin, Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations equations under large perturbation,, J. Differential Equations, 246 (2009), 4077.  doi: 10.1016/j.jde.2009.01.017.  Google Scholar [12] F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motion,, Adv. Math., 219 (2008), 1246.  doi: 10.1016/j.aim.2008.06.014.  Google Scholar [13] J. Kanel, On a model system of equations of one-dimensional gas motion,, Differencial nye Uravnenija, 4 (1968), 721.   Google Scholar [14] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, Comm. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar [15] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas,, Proc. Japan Acad. Ser.-A, 62 (1986), 249.  doi: 10.3792/pjaa.62.249.  Google Scholar [16] S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space,, Comm. Math. Phys., 240 (2003), 483.  doi: 10.1007/s00220-003-0909-2.  Google Scholar [17] S. Kawashima and P. Zhu, Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space,, J. Differential Equations, 244 (2008), 3151.  doi: 10.1016/j.jde.2008.01.020.  Google Scholar [18] H.-L. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^3$,, Arch. Ration. Mech. Anal., 196 (2010), 681.  doi: 10.1007/s00205-009-0255-4.  Google Scholar [19] H.-L. Li, T. Yang and C. Zou, Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system,, Acta Math. Sci.-B, 29 (2009), 1721.  doi: 10.1016/S0252-9602(10)60013-6.  Google Scholar [20] T.-P. Liu and Z.-P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations,, Comm. Math. Phys., 118 (1988), 451.   Google Scholar [21] A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary,, Arch. Ration. Mech. Anal., 146 (1999), 1.  doi: 10.1007/s002050050134.  Google Scholar [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.   Google Scholar [23] A. Matsumura and T. Nishida, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 3 (1986), 1.  doi: 10.1007/BF03167088.  Google Scholar [24] A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves for solutions of viscous p-system with boundary effect,, Quart. Appl. Math., 58 (2000), 69.   Google Scholar [25] A. Matsumura and K. Nishihara, Large-time behavior of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas,, Comm. Math. Phys., 222 (2001), 449.  doi: 10.1007/s002200100517.  Google Scholar [26] D. R. Nicholson, "Introduction to Plasma Theory,", Wiley, (1983).   Google Scholar [27] K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations,, SIAM J. Math. Anal., 35 (2004), 1561.  doi: 10.1137/S003614100342735X.  Google Scholar [28] S. Wang and S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 31 (2006), 571.  doi: 10.1080/03605300500361487.  Google Scholar [29] W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions,, J. Differential Equations, 248 (2010), 1617.  doi: 10.1016/j.jde.2010.01.003.  Google Scholar [30] G. Zhang, H.-L. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^3$,, J. Differential Equations, 250 (2011), 866.  doi: 10.1016/j.jde.2010.07.035.  Google Scholar

show all references

##### References:
 [1] C. Besse, J. Claudel and P. Degond, et al., A model hierarchy for ionospheric plasma modeling,, Math. Models Methods Appl. Sci., 14 (2004), 393.  doi: 10.1142/S0218202504003283.  Google Scholar [2] D. Chae, On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in $R^N$,, Comm. Partial Differential Equations, 35 (2010), 535.  doi: 10.1080/03605300903473418.  Google Scholar [3] S. Cordier, P. Degond, P. Markowich and C. Schmeiser, Travelling wave analysis of an isothermal Euler-Poisson model,, Ann. Fac. Sci. Toulouse Math., 5 (1996), 599.  doi: 10.1.1.57.5963.  Google Scholar [4] D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem,, Quart. Appl. Math., 61 (2003), 345.  doi: 10.1.1.14.6956.  Google Scholar [5] D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data,, Nonlinearity, 21 (2008), 135.  doi: 10.1088/0951-7715/21/1/008.  Google Scholar [6] R.-J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system,, Analysis and Applications, 10 (2012), 133.  doi: 10.1142/S0219530512500078.  Google Scholar [7] R.-J. Duan, Q. Q. Liu and C. J. Zhu, The Cauchy problem on the compressible two-fluids Euler-Maxwell equations,, SIAM J. Math. Anal., 44 (2012), 102.  doi: 10.1137/110838406.  Google Scholar [8] C. Hao and H.-L. Li, Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions,, J. Differential Equations, 246 (2009), 4791.  doi: 10.1016/j.jde.2009.09.008.  Google Scholar [9] L. Hsiao and H.-L. Li, Compressible Navier-Stokes-Poisson equations,, Acta Math Sci-B, 30 (2010), 1937.  doi: 10.1016/S0252-9602(10)60184-1.  Google Scholar [10] F. Huang, J. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system,, Arch. Ration. Mech. Anal., 197 (2010), 89.  doi: 10.1007/s00205-009-0267-0.  Google Scholar [11] F. Huang and X. Qin, Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations equations under large perturbation,, J. Differential Equations, 246 (2009), 4077.  doi: 10.1016/j.jde.2009.01.017.  Google Scholar [12] F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motion,, Adv. Math., 219 (2008), 1246.  doi: 10.1016/j.aim.2008.06.014.  Google Scholar [13] J. Kanel, On a model system of equations of one-dimensional gas motion,, Differencial nye Uravnenija, 4 (1968), 721.   Google Scholar [14] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion,, Comm. Math. Phys., 101 (1985), 97.  doi: 10.1007/BF01212358.  Google Scholar [15] S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas,, Proc. Japan Acad. Ser.-A, 62 (1986), 249.  doi: 10.3792/pjaa.62.249.  Google Scholar [16] S. Kawashima, S. Nishibata and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space,, Comm. Math. Phys., 240 (2003), 483.  doi: 10.1007/s00220-003-0909-2.  Google Scholar [17] S. Kawashima and P. Zhu, Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space,, J. Differential Equations, 244 (2008), 3151.  doi: 10.1016/j.jde.2008.01.020.  Google Scholar [18] H.-L. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^3$,, Arch. Ration. Mech. Anal., 196 (2010), 681.  doi: 10.1007/s00205-009-0255-4.  Google Scholar [19] H.-L. Li, T. Yang and C. Zou, Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system,, Acta Math. Sci.-B, 29 (2009), 1721.  doi: 10.1016/S0252-9602(10)60013-6.  Google Scholar [20] T.-P. Liu and Z.-P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations,, Comm. Math. Phys., 118 (1988), 451.   Google Scholar [21] A. Matsumura and M. Mei, Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary,, Arch. Ration. Mech. Anal., 146 (1999), 1.  doi: 10.1007/s002050050134.  Google Scholar [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases,, J. Math. Kyoto Univ., 20 (1980), 67.   Google Scholar [23] A. Matsumura and T. Nishida, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 3 (1986), 1.  doi: 10.1007/BF03167088.  Google Scholar [24] A. Matsumura and K. Nishihara, Asymptotic toward the rarefaction waves for solutions of viscous p-system with boundary effect,, Quart. Appl. Math., 58 (2000), 69.   Google Scholar [25] A. Matsumura and K. Nishihara, Large-time behavior of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas,, Comm. Math. Phys., 222 (2001), 449.  doi: 10.1007/s002200100517.  Google Scholar [26] D. R. Nicholson, "Introduction to Plasma Theory,", Wiley, (1983).   Google Scholar [27] K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations,, SIAM J. Math. Anal., 35 (2004), 1561.  doi: 10.1137/S003614100342735X.  Google Scholar [28] S. Wang and S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 31 (2006), 571.  doi: 10.1080/03605300500361487.  Google Scholar [29] W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions,, J. Differential Equations, 248 (2010), 1617.  doi: 10.1016/j.jde.2010.01.003.  Google Scholar [30] G. Zhang, H.-L. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^3$,, J. Differential Equations, 250 (2011), 866.  doi: 10.1016/j.jde.2010.07.035.  Google Scholar
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