# American Institute of Mathematical Sciences

September  2016, 36(9): 4839-4870. doi: 10.3934/dcds.2016009

## Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate

 1 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China, China, China, China

Received  June 2015 Revised  January 2016 Published  May 2016

In this paper, we study the asymptotic behavior of solution to the initial boundary value problem for the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line $\mathbb{R}_{+}:=(0,\infty).$ Our idea mainly comes from [10] which describes the large time behavior of solution for the non-isentropic Navier-Stokes equations in a half line. The electric field brings us some additional troubles compared with Navier-Stokes equations in the absence of the electric field. We obtain the convergence rate of global solution towards corresponding stationary solution. 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 proofs are given by a weighted energy method.
Citation: Haibo Cui, Zhensheng Gao, Haiyan Yin, Peixing Zhang. Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
##### References:
 [1] J. Carr, Applications of Centre Manifold Theory, Springer Verlag, 1981. [2] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Second edition, Plenum Press, 1984. [3] D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math., 61 (2003), 345-361. [4] R. J. Duan and S. Q. Liu, Stability of rarefaction waves of the Navier-Stokes-Poisson system, J. Differential Equations, 258 (2015), 2495-2530. doi: 10.1016/j.jde.2014.12.019. [5] R. J. Duan and S. Q. Liu, Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 47 (2015), 3585-3647. doi: 10.1137/140995179. [6] R. J. Duan, S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84. doi: 10.1007/s11425-015-5059-4. [7] R. J. Duan and X. F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Comm. Pure Appl. Anal., 12 (2013), 985-1014. doi: 10.3934/cpaa.2013.12.985. [8] F. M. Huang and X. H. Qin, Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation, J. Differential Equations, 246 (2009), 4077-4096. doi: 10.1016/j.jde.2009.01.017. [9] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. [10] S. Kawashima, T. Nakamura, S. Nishibata and P. C. Zhu, Stationary waves to viscous heat-conductive gases in half space: Existence, stability and convergence rate, Math. Models Methods Appl. Sci., 20 (2010), 2201-2035. doi: 10.1142/S0218202510004908. [11] S. Kawashima, S. Nishibata and P. C. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2. [12] H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$, Arch. Ration. Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4. [13] S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of contact discontinuity for the Navier-Stokes-Poisson system with free boundary, preprint, arXiv:1508.01405. [14] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, New York, 1990. doi: 10.1007/978-3-7091-6961-2. [15] 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), 1-22. doi: 10.1007/s002050050134. [16] A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), 449-474. doi: 10.1007/s002200100517. [17] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016. [18] T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, Journal of Hyperbolic Differential Equations, 8 (2011), 651-670. doi: 10.1142/S0219891611002524. [19] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132. [20] L. Z. Ruan, H. Y. Yin and C. J. Zhu, The stability of the superposition of rarefaction wave and contact discontinuity for the Navier-Stokes-Poisson system with free boundary, preprint. [21] Z. Tan, T. Yang, H. J. Zhao and Q. Y. Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal., 45 (2013), 547-571. doi: 10.1137/120876174. [22] H. Y. Yin, J. S. Zhang and C. J. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Analysis: Real World Applications, 31 (2016), 492-512, arXiv:1508.01411. doi: 10.1016/j.nonrwa.2016.01.020. [23] G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$, J.Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035. [24] F. Zhou and Y. P. Li, Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line, Bound. Value Probl., 2013 (2013), 1-22. doi: 10.1186/1687-2770-2013-124.

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##### References:
 [1] J. Carr, Applications of Centre Manifold Theory, Springer Verlag, 1981. [2] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Second edition, Plenum Press, 1984. [3] D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math., 61 (2003), 345-361. [4] R. J. Duan and S. Q. Liu, Stability of rarefaction waves of the Navier-Stokes-Poisson system, J. Differential Equations, 258 (2015), 2495-2530. doi: 10.1016/j.jde.2014.12.019. [5] R. J. Duan and S. Q. Liu, Stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 47 (2015), 3585-3647. doi: 10.1137/140995179. [6] R. J. Duan, S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84. doi: 10.1007/s11425-015-5059-4. [7] R. J. Duan and X. F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Comm. Pure Appl. Anal., 12 (2013), 985-1014. doi: 10.3934/cpaa.2013.12.985. [8] F. M. Huang and X. H. Qin, Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation, J. Differential Equations, 246 (2009), 4077-4096. doi: 10.1016/j.jde.2009.01.017. [9] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. [10] S. Kawashima, T. Nakamura, S. Nishibata and P. C. Zhu, Stationary waves to viscous heat-conductive gases in half space: Existence, stability and convergence rate, Math. Models Methods Appl. Sci., 20 (2010), 2201-2035. doi: 10.1142/S0218202510004908. [11] S. Kawashima, S. Nishibata and P. C. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500. doi: 10.1007/s00220-003-0909-2. [12] H. L. Li, A. Matsumura and G. J. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$, Arch. Ration. Mech. Anal., 196 (2010), 681-713. doi: 10.1007/s00205-009-0255-4. [13] S. Q. Liu, H. Y. Yin and C. J. Zhu, Stability of contact discontinuity for the Navier-Stokes-Poisson system with free boundary, preprint, arXiv:1508.01405. [14] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer, New York, 1990. doi: 10.1007/978-3-7091-6961-2. [15] 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), 1-22. doi: 10.1007/s002050050134. [16] A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), 449-474. doi: 10.1007/s002200100517. [17] T. Nakamura, S. Nishibata and T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111. doi: 10.1016/j.jde.2007.06.016. [18] T. Nakamura and S. Nishibata, Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, Journal of Hyperbolic Differential Equations, 8 (2011), 651-670. doi: 10.1142/S0219891611002524. [19] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac., 41 (1998), 107-132. [20] L. Z. Ruan, H. Y. Yin and C. J. Zhu, The stability of the superposition of rarefaction wave and contact discontinuity for the Navier-Stokes-Poisson system with free boundary, preprint. [21] Z. Tan, T. Yang, H. J. Zhao and Q. Y. Zou, Global solutions to the one-dimensional compressible Navier-Stokes-Poisson equations with large data, SIAM J. Math. Anal., 45 (2013), 547-571. doi: 10.1137/120876174. [22] H. Y. Yin, J. S. Zhang and C. J. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Analysis: Real World Applications, 31 (2016), 492-512, arXiv:1508.01411. doi: 10.1016/j.nonrwa.2016.01.020. [23] G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^{3}$, J.Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035. [24] F. Zhou and Y. P. Li, Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line, Bound. Value Probl., 2013 (2013), 1-22. doi: 10.1186/1687-2770-2013-124.
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