American Institute of Mathematical Sciences

Advanced
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

Haibo Cui 1, , Zhensheng Gao 1, , Haiyan Yin 1, and  Peixing Zhang 1,

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.
Keywords: outflow problem, stationary solution, Navier-Stokes-Poisson system, weighted energy method., convergence rate.
Mathematics Subject Classification: Primary: 35M33, 35B35; Secondary: 35B4.
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 & Continuous Dynamical Systems - A, 2016, 36 (9) : 4839-4870. doi: 10.3934/dcds.2016009
References:
[1]

J. Carr, Applications of Centre Manifold Theory,, Springer Verlag, (1981).   Google Scholar

[2]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Second edition, (1984).   Google Scholar

[3]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem,, Quart. Appl. Math., 61 (2003), 345.   Google Scholar

[4]

R. J. Duan and S. Q. Liu, Stability of rarefaction waves of the Navier-Stokes-Poisson system,, J. Differential Equations, 258 (2015), 2495.  doi: 10.1016/j.jde.2014.12.019.  Google Scholar

[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.  doi: 10.1137/140995179.  Google Scholar

[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.  doi: 10.1007/s11425-015-5059-4.  Google Scholar

[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.  doi: 10.3934/cpaa.2013.12.985.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.01.017.  Google Scholar

[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.  doi: 10.1007/BF01212358.  Google Scholar

[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.  doi: 10.1142/S0218202510004908.  Google Scholar

[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.  doi: 10.1007/s00220-003-0909-2.  Google Scholar

[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.  doi: 10.1007/s00205-009-0255-4.  Google Scholar

[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, ().   Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.  doi: 10.1007/s002050050134.  Google Scholar

[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.  doi: 10.1007/s002200100517.  Google Scholar

[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.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar

[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.  doi: 10.1142/S0219891611002524.  Google Scholar

[19]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, Funkcial. Ekvac., 41 (1998), 107.   Google Scholar

[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., ().   Google Scholar

[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.  doi: 10.1137/120876174.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2016.01.020.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.07.035.  Google Scholar

[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.  doi: 10.1186/1687-2770-2013-124.  Google Scholar

show all references

References:
[1]

J. Carr, Applications of Centre Manifold Theory,, Springer Verlag, (1981).   Google Scholar

[2]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Second edition, (1984).   Google Scholar

[3]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem,, Quart. Appl. Math., 61 (2003), 345.   Google Scholar

[4]

R. J. Duan and S. Q. Liu, Stability of rarefaction waves of the Navier-Stokes-Poisson system,, J. Differential Equations, 258 (2015), 2495.  doi: 10.1016/j.jde.2014.12.019.  Google Scholar

[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.  doi: 10.1137/140995179.  Google Scholar

[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.  doi: 10.1007/s11425-015-5059-4.  Google Scholar

[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.  doi: 10.3934/cpaa.2013.12.985.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.01.017.  Google Scholar

[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.  doi: 10.1007/BF01212358.  Google Scholar

[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.  doi: 10.1142/S0218202510004908.  Google Scholar

[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.  doi: 10.1007/s00220-003-0909-2.  Google Scholar

[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.  doi: 10.1007/s00205-009-0255-4.  Google Scholar

[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, ().   Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[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.  doi: 10.1007/s002050050134.  Google Scholar

[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.  doi: 10.1007/s002200100517.  Google Scholar

[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.  doi: 10.1016/j.jde.2007.06.016.  Google Scholar

[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.  doi: 10.1142/S0219891611002524.  Google Scholar

[19]

M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws,, Funkcial. Ekvac., 41 (1998), 107.   Google Scholar

[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., ().   Google Scholar

[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.  doi: 10.1137/120876174.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2016.01.020.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.07.035.  Google Scholar

[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.  doi: 10.1186/1687-2770-2013-124.  Google Scholar

[1]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[2]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020210
[3]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030
[4]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[5]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[6]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[7]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[8]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185
[9]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
[10]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[11]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[12]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[13]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028
[14]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[15]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[16]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
[17]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[18]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511
[19]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[20]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences