# American Institute of Mathematical Sciences

February  2016, 36(2): 611-629. doi: 10.3934/dcds.2016.36.611

## Time periodic solutions to Navier-Stokes-Korteweg system with friction

 1 School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005 2 School of Mathematical Sciences and Fujian Provincial Key Laboratory, on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, 361005

Received  December 2013 Revised  March 2015 Published  August 2015

In this paper, the compressible Navier-Stokes-Korteweg system with friction is considered in $\mathbb{R}^3$. Via the linear analysis, we show the existence, uniqueness and time-asymptotic stability of the time periodic solution when a time periodic external force is taken into account. Our proof is based on a combination of the energy method and the contraction mapping theorem. In particular, this is the first paper that a time periodic solution can be obtained in the whole space $\mathbb{R}^3$ only under the suitable smallness condition of $H^{N-1}\cap L^1$--norm$(N\geq5)$ of time periodic external force.
Citation: Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions to Navier-Stokes-Korteweg system with friction. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 611-629. doi: 10.3934/dcds.2016.36.611
##### References:
 [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] 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. [3] 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. [4] Z. Z. Chen, Q. H. Xiao and H. J. Zhao, Time periodic solutions of compressible fluid models of Korteweg type, it Math.Phys., Preprint, arXiv:1203.6529 (2012). [5] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri 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 (2012). 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. Henri 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 problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [16] 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. [17] M. E. Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2. [18] S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dynam. Systems, 14 (2006), 579-596. doi: 10.3934/dcds.2006.14.579. [19] 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. [20] 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. [21] 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.

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##### References:
 [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] 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. [3] 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. [4] Z. Z. Chen, Q. H. Xiao and H. J. Zhao, Time periodic solutions of compressible fluid models of Korteweg type, it Math.Phys., Preprint, arXiv:1203.6529 (2012). [5] R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri 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 (2012). 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. Henri 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 problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104. [16] 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. [17] M. E. Taylor, Partial Differential Equations III, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-4187-2. [18] S. Ukai, Time periodic solutions of Boltzmann equation, Discrete Contin. Dynam. Systems, 14 (2006), 579-596. doi: 10.3934/dcds.2006.14.579. [19] 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. [20] 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. [21] 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.
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