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

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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

Abstract
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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 and 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.
[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.
[3]	Z. Z. Chen, Q. H. Xiao and H. J. Zhao, Time periodic solutions of compressible fluid models of Korteweg type, preprint, arXiv:1203.6529.
[4]	R. Danchi, Global existence in critical space for compressible Navier-Stokes equations, Invent.math., 141 (2000), 579-614. doi: 10.1007/s002220000078.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	S. Kawashima, Systems of A Hyperbolic-parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	Michael E. Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.
[19]	S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dyn. Syst., 14 (2006), 579-596. doi: 10.3934/dcds.2006.14.579.
[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.
[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.
[22]	G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Journal of Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035.
[23]	X. Zhang and Z. Tan, Decay estimates of the non-isentropic compressible fluid models of Korteweg type in $\mathbb{R}^3$, Comm. Math. Sci., 12 (2014), 1437-1456. doi: 10.4310/CMS.2014.v12.n8.a4.

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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.
[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.
[3]	Z. Z. Chen, Q. H. Xiao and H. J. Zhao, Time periodic solutions of compressible fluid models of Korteweg type, preprint, arXiv:1203.6529.
[4]	R. Danchi, Global existence in critical space for compressible Navier-Stokes equations, Invent.math., 141 (2000), 579-614. doi: 10.1007/s002220000078.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	S. Kawashima, Systems of A Hyperbolic-parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	Michael E. Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2.
[19]	S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dyn. Syst., 14 (2006), 579-596. doi: 10.3934/dcds.2006.14.579.
[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.
[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.
[22]	G. J. Zhang, H. L. Li and C. J. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Journal of Differential Equations, 250 (2011), 866-891. doi: 10.1016/j.jde.2010.07.035.
[23]	X. Zhang and Z. Tan, Decay estimates of the non-isentropic compressible fluid models of Korteweg type in $\mathbb{R}^3$, Comm. Math. Sci., 12 (2014), 1437-1456. doi: 10.4310/CMS.2014.v12.n8.a4.

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