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 and 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), 393-415. doi: 10.1142/S0218202504003283.

[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), 535-557. doi: 10.1080/03605300903473418.

[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), 599-643. doi: 10.1.1.57.5963.

[4]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math., 61 (2003), 345-361. doi: 10.1.1.14.6956.

[5]

D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data, Nonlinearity, 21 (2008), 135-148. doi: 10.1088/0951-7715/21/1/008.

[6]

R.-J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Analysis and Applications, 10 (2012), 133-197. doi: 10.1142/S0219530512500078.

[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), 102-133. doi: 10.1137/110838406.

[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), 4791-4812. doi: 10.1016/j.jde.2009.09.008.

[9]

L. Hsiao and H.-L. Li, Compressible Navier-Stokes-Poisson equations, Acta Math Sci-B, 30 (2010), 1937-1948. doi: 10.1016/S0252-9602(10)60184-1.

[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), 89-116. doi: 10.1007/s00205-009-0267-0.

[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), 4077-4096. doi: 10.1016/j.jde.2009.01.017.

[12]

F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motion, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014.

[13]

J. Kanel, On a model system of equations of one-dimensional gas motion, Differencial nye Uravnenija, 4 (1968), 721-734.

[14]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[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), 249-252. doi: 10.3792/pjaa.62.249.

[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), 483-500. doi: 10.1007/s00220-003-0909-2.

[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), 3151-3179. doi: 10.1016/j.jde.2008.01.020.

[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), 681-713. doi: 10.1007/s00205-009-0255-4.

[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), 1721-1736. doi: 10.1016/S0252-9602(10)60013-6.

[20]

T.-P. Liu and Z.-P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys.,118 (1988), 451-465.

[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), 1-22. doi: 10.1007/s002050050134.

[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), 67-104.

[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), 1-13. doi: 10.1007/BF03167088.

[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), 69-83.

[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), 449-474. doi: 10.1007/s002200100517.

[26]

D. R. Nicholson, "Introduction to Plasma Theory," Wiley, 1983.

[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), 1561-1597. doi: 10.1137/S003614100342735X.

[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), 571-591. doi: 10.1080/03605300500361487.

[29]

W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differential Equations, 248 (2010), 1617-1636. doi: 10.1016/j.jde.2010.01.003.

[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), 866-891. doi: 10.1016/j.jde.2010.07.035.

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), 393-415. doi: 10.1142/S0218202504003283.

[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), 535-557. doi: 10.1080/03605300903473418.

[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), 599-643. doi: 10.1.1.57.5963.

[4]

D. Donatelli, Local and global existence for the coupled Navier-Stokes-Poisson problem, Quart. Appl. Math., 61 (2003), 345-361. doi: 10.1.1.14.6956.

[5]

D. Donatelli and P. Marcati, A quasineutral type limit for the Navier-Stokes-Poisson system with large data, Nonlinearity, 21 (2008), 135-148. doi: 10.1088/0951-7715/21/1/008.

[6]

R.-J. Duan, Green's function and large time behavior of the Navier-Stokes-Maxwell system, Analysis and Applications, 10 (2012), 133-197. doi: 10.1142/S0219530512500078.

[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), 102-133. doi: 10.1137/110838406.

[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), 4791-4812. doi: 10.1016/j.jde.2009.09.008.

[9]

L. Hsiao and H.-L. Li, Compressible Navier-Stokes-Poisson equations, Acta Math Sci-B, 30 (2010), 1937-1948. doi: 10.1016/S0252-9602(10)60184-1.

[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), 89-116. doi: 10.1007/s00205-009-0267-0.

[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), 4077-4096. doi: 10.1016/j.jde.2009.01.017.

[12]

F. Huang, Z. Xin and T. Yang, Contact discontinuity with general perturbations for gas motion, Adv. Math., 219 (2008), 1246-1297. doi: 10.1016/j.aim.2008.06.014.

[13]

J. Kanel, On a model system of equations of one-dimensional gas motion, Differencial nye Uravnenija, 4 (1968), 721-734.

[14]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358.

[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), 249-252. doi: 10.3792/pjaa.62.249.

[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), 483-500. doi: 10.1007/s00220-003-0909-2.

[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), 3151-3179. doi: 10.1016/j.jde.2008.01.020.

[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), 681-713. doi: 10.1007/s00205-009-0255-4.

[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), 1721-1736. doi: 10.1016/S0252-9602(10)60013-6.

[20]

T.-P. Liu and Z.-P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys.,118 (1988), 451-465.

[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), 1-22. doi: 10.1007/s002050050134.

[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), 67-104.

[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), 1-13. doi: 10.1007/BF03167088.

[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), 69-83.

[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), 449-474. doi: 10.1007/s002200100517.

[26]

D. R. Nicholson, "Introduction to Plasma Theory," Wiley, 1983.

[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), 1561-1597. doi: 10.1137/S003614100342735X.

[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), 571-591. doi: 10.1080/03605300500361487.

[29]

W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differential Equations, 248 (2010), 1617-1636. doi: 10.1016/j.jde.2010.01.003.

[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), 866-891. doi: 10.1016/j.jde.2010.07.035.

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