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March  2015, 8(1): 29-51. doi: 10.3934/krm.2015.8.29

Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type

Hong Cai 1, , Zhong Tan 1, and  Qiuju Xu 1,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China, China

Received  August 2013 Revised  October 2013 Published  December 2014

In this paper, the non-isentropic compressible Navier-Stokes-Korteweg system with a time periodic external force is considered in $\mathbb{R}^n$. The optimal time decay rates are obtained by spectral analysis. Using the optimal decay estimates, we show that the existence, uniqueness and time-asymptotic stability of time periodic solutions when the space dimension $n\geq 5$. Our proof is based on a combination of the energy method and the contraction mapping theorem.
Keywords: optimal time decay rates, energy estimates., Non-isentropic compressible Navier-Stokes-Korteweg system, time periodic solution.
Mathematics Subject Classification: Primary: 35M10, 35Q35; Secondary: 35B1.
Citation: Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinetic & Related Models, 2015, 8 (1) : 29-51. doi: 10.3934/krm.2015.8.29
References:
[1]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.  Google Scholar

[2]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamics equations, Nonlinear Anal., 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019.  Google Scholar

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Z. Z. Chen, Q. H. Xiao and H. J. Zhao, Time periodic solutions of compressible fluid models of Korteweg type,, preprint, ().   Google Scholar

[4]

R. Danchi, Global existence in critical space for compressible Navier-Stokes equations, Invent.math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[5]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincare Anal. Nonlinear, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1.  Google Scholar

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R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal decay estimates on the linearized Boltzmann equations with time dependent force and their applications, Comm. Math. Phys., 277 (2008), 189-236. doi: 10.1007/s00220-007-0366-4.  Google Scholar

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B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249. doi: 10.1007/s00021-009-0013-2.  Google Scholar

[8]

B. Haspot, Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension, Methods Appl. Anal., 20 (2013), 141-164. arXiv:1211.4819. doi: 10.4310/MAA.2013.v20.n2.a3.  Google Scholar

[9]

H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.  Google Scholar

[10]

H. Hattori and D. Li, Golobal solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.  Google Scholar

[11]

S. Kawashima, Systems of A Hyperbolic-parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983. Google Scholar

[12]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincare Anal. Nonlinear, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005.  Google Scholar

[13]

Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232. doi: 10.1016/j.jmaa.2011.11.006.  Google Scholar

[14]

H. F. Ma, S. Ukai and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293. doi: 10.1016/j.jde.2009.11.031.  Google Scholar

[15]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fuids, Proc. Japan Acad. Ser. A, 55 (1979) , 337-342. doi: 10.3792/pjaa.55.337.  Google Scholar

[16]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[17]

Z. Tan and H. Q. wang, Time periodic solutions of compressible magnetohydrodynamic equations, Nonlinear Anal., 76 (2013), 153-164. doi: 10.1016/j.na.2012.08.012.  Google Scholar

[18]

Michael E. Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.  Google Scholar

[19]

S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dyn. Syst., 14 (2006), 579-596. doi: 10.3934/dcds.2006.14.579.  Google Scholar

[20]

S. Ukai and T. Yang, The Boltzmann equation in the sapce $L^2\cap L^{\infty}_\beta$: global and time periodic solution, Analysis and Applications, 4 (2006), 263-310. doi: 10.1142/S0219530506000784.  Google Scholar

[21]

Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006.  Google Scholar

[22]

G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^3$, Journal of Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035.  Google Scholar

[23]

X. Zhang and Z. Tan, Decay estimates of the non-isentropic compressible fluid models of Korteweg type in $\mathbbR^3$, Comm. Math. Sci., 12 (2014), 1437-1456. doi: 10.4310/CMS.2014.v12.n8.a4.  Google Scholar

show all references

References:
[1]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.  Google Scholar

[2]

Q. Chen and Z. Tan, Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamics equations, Nonlinear Anal., 72 (2010), 4438-4451. doi: 10.1016/j.na.2010.02.019.  Google Scholar

[3]

Z. Z. Chen, Q. H. Xiao and H. J. Zhao, Time periodic solutions of compressible fluid models of Korteweg type,, preprint, ().   Google Scholar

[4]

R. Danchi, Global existence in critical space for compressible Navier-Stokes equations, Invent.math., 141 (2000), 579-614. doi: 10.1007/s002220000078.  Google Scholar

[5]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincare Anal. Nonlinear, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1.  Google Scholar

[6]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal decay estimates on the linearized Boltzmann equations with time dependent force and their applications, Comm. Math. Phys., 277 (2008), 189-236. doi: 10.1007/s00220-007-0366-4.  Google Scholar

[7]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249. doi: 10.1007/s00021-009-0013-2.  Google Scholar

[8]

B. Haspot, Existence of global strong solution for the compressible Navier-Stokes system and the Korteweg system in two-dimension, Methods Appl. Anal., 20 (2013), 141-164. arXiv:1211.4819. doi: 10.4310/MAA.2013.v20.n2.a3.  Google Scholar

[9]

H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.  Google Scholar

[10]

H. Hattori and D. Li, Golobal solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.  Google Scholar

[11]

S. Kawashima, Systems of A Hyperbolic-parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983. Google Scholar

[12]

M. Kotschote, Strong solutions for a compressible fluid model of Korteweg type, Ann. Inst. H. Poincare Anal. Nonlinear, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005.  Google Scholar

[13]

Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232. doi: 10.1016/j.jmaa.2011.11.006.  Google Scholar

[14]

H. F. Ma, S. Ukai and T. Yang, Time periodic solutions of compressible Navier-Stokes equations, J. Differential Equations, 248 (2010), 2275-2293. doi: 10.1016/j.jde.2009.11.031.  Google Scholar

[15]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fuids, Proc. Japan Acad. Ser. A, 55 (1979) , 337-342. doi: 10.3792/pjaa.55.337.  Google Scholar

[16]

A. Matsumura and T. Nishida, The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[17]

Z. Tan and H. Q. wang, Time periodic solutions of compressible magnetohydrodynamic equations, Nonlinear Anal., 76 (2013), 153-164. doi: 10.1016/j.na.2012.08.012.  Google Scholar

[18]

Michael E. Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.  Google Scholar

[19]

S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dyn. Syst., 14 (2006), 579-596. doi: 10.3934/dcds.2006.14.579.  Google Scholar

[20]

S. Ukai and T. Yang, The Boltzmann equation in the sapce $L^2\cap L^{\infty}_\beta$: global and time periodic solution, Analysis and Applications, 4 (2006), 263-310. doi: 10.1142/S0219530506000784.  Google Scholar

[21]

Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006.  Google Scholar

[22]

G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbbR^3$, Journal of Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035.  Google Scholar

[23]

X. Zhang and Z. Tan, Decay estimates of the non-isentropic compressible fluid models of Korteweg type in $\mathbbR^3$, Comm. Math. Sci., 12 (2014), 1437-1456. doi: 10.4310/CMS.2014.v12.n8.a4.  Google Scholar

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