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Time-Periodic Solution to the Compressible Navier-Stokes/Allen-Cahn System

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Abstract

In this paper, we investigate the time-periodic solution to a coupled compressible Navier- Stokes/Allen-Cahn system which describes the motion of a mixture of two viscous compressible fluids with a time periodic external force in a periodic domain in ℝN. The existence of the time-periodic solution to the system is established by using an approach of parabolic regularization and combining with the topology degree theory, and then the uniqueness of the period solution is obtained under some smallness and symmetry assumptions on the external force.

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References

  1. Abels, H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Ration. Mech. Anal., 194, 463–506 (2009)

    Article  MathSciNet  Google Scholar 

  2. Abels, H., Feireisl, E.: On a diffuse interface model for a two-phase flow of viscous fluids. Indiana Univ. Math. J., 57(2), 659–698 (2008)

    Article  MathSciNet  Google Scholar 

  3. Blesgen, T.: A generalization of the Navier-Stokes equations to two-phase flows. J. Phys. D: Appl. Phys., 32, 1119–1123 (1999)

    Article  Google Scholar 

  4. Boyer, F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal., 20(2), 175–212 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Boyer, F.: A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids, 31, 41–68 (2002)

    Article  Google Scholar 

  6. Cai, H., Tan, Z., Xu, Q.: Time periodic solutions to Navier-Stokes-Korteweg system with friction. Discrete Contin. Dyn. Syst., 36(2), 611–629 (2016)

    Article  MathSciNet  Google Scholar 

  7. Chen, M., Guo, X.: Global large solutions for a coupled compressible Navier-Stokes-Allen-Cahn system with initial vacuum. Nonlinear Anal. Real World Appl., 37, 350–373 (2017)

    Article  MathSciNet  Google Scholar 

  8. Fan, J., Yu, W.: Global variational solutions to the compressible magnetohydrodynamic equations. Nonlinear Anal., 69, 3637–3660 (2008)

    Article  MathSciNet  Google Scholar 

  9. Feireisl, E., Petzeltová, H., Rocca, E., et al.: Analysis of a phase-field model for two-phase compressible fluids. Math. Models Methods Appl. Sci., 7, 1129–1160 (2010)

    Article  MathSciNet  Google Scholar 

  10. Gal, C. G., Grasselli, M.: Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. Ann. I. H. Poincaré, 27, 401–436 (2010)

    Article  MathSciNet  Google Scholar 

  11. Gal, C. G., Grasselli, M.: Instability of two-phase flows: a lower bound on the dimension of global attractor of the Cahn-Hilliard-Navier-Stokes system. Physica D, 240, 629–635 (2011)

    Article  MathSciNet  Google Scholar 

  12. Hosseini, B. S., Turek, S., Möller, M., et al.: Isogeometric analysis of the Navier-Stokes-Cahn-Hilliard equations with application to incompressible two-phase flows. J. Comput. Phys., 348, 171–194 (2017)

    Article  MathSciNet  Google Scholar 

  13. Jin, C., Yang, T.: Time periodic solution for a 3-D compressible Navier-Stokes system with an external force in ℝ3. J. Differential Equations, 259(7), 2576–2601 (2015)

    Article  MathSciNet  Google Scholar 

  14. Jin, C.: Periodic solution for a non-isentropic compressible Navier-Stokes equations in a bounded domain. J. Math. Phys., 56(4), 35–92 (2015)

    Article  MathSciNet  Google Scholar 

  15. Jin, C., Yang, T.: Time periodic solution to the compressible Navier-Stokes equations in a periodic domain. Acta Math. Sci. Ser. B, English Ed., 36(4), 1015–1029 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kagei, Y., Tsuda, K.: Existence and stability of time periodic solution to the compressible Navier-Stokes equation for time periodic external force with symmetry. J. Differential Equations, 258, 399–444 (2015)

    Article  MathSciNet  Google Scholar 

  17. Kotschote, M., Zacher, R.: Strong solutions in the dynamical theory of compressible fluid mixtures. Math. Models Methods Appl. Sci., 25(7), 1217–1256 (2015)

    Article  MathSciNet  Google Scholar 

  18. Liu, W., Bertozzi, A. L., Kolokolnikov, T.: Diffuse interface surface tension models in an expanding flow. Commun. Math. Sci., 10(1), 387–418 (2012)

    Article  MathSciNet  Google Scholar 

  19. Ma, H., Ukai, P., Yang, T.: Time periodic solutions of compressible Navier-Stokes equations. J. Differential Equations, 248, 2275–2293 (2010)

    Article  MathSciNet  Google Scholar 

  20. Medjo T. T.: On the existence and uniqueness of solution to a stochastic 2D Navier-Stokes-Cahn-Hilliard model. J. Differential Equations, 263, 1028–1054 (2017)

    Article  MathSciNet  Google Scholar 

  21. Tan, Z., Lim, K. M., Khoo, B. C.: An adaptive mesh redistribution method for the incompressible mixture flows using phase-field model. J. Comput. Phys., 225, 1137–1158 (2007)

    Article  MathSciNet  Google Scholar 

  22. Taylor, M. E.: Partial Differential Equations III, Springer-Verlag, New York, 1996

    Book  Google Scholar 

  23. Witterstein, G.: On the existence of a solution for a model of stem cell differentiation. Math. Methods Appl. Sci., 31, 1972–1995 (2008)

    Article  MathSciNet  Google Scholar 

  24. Yang, X., Feng, J. J., Liu, C., et al.: Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys., 218, 417–428 (2006)

    Article  MathSciNet  Google Scholar 

  25. Zhang J.: A regularity criterion for the 3D incompressible density-dependent Navier-Stokes-Allen-Cahn equations. J. Partial Differential Equations, 2, 116–123 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Zhang, J.: Regularity of solutions to 1D compressible Navier-Stokes-Allen-Cahn system. Appl. Anal., DOI: 10.1080/00036811.2019.1668554 (2019)

    Google Scholar 

  27. Zhao, L., Guo, B., Huang, H.: Vanishing viscosity limit for a coupled Navier-Stokes/Allen-Cahn system. J. Math. Anal. Appl., 384, 232–245 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We thank the referees for their time and valuable comments.

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, Zhongyuan University of Technology, Zhengzhou, 450007, P. R. China

    Chang Ming Song, Jian Lin Zhang & Yuan Yuan Wang

Authors
  1. Chang Ming Song
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  2. Jian Lin Zhang
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  3. Yuan Yuan Wang
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Corresponding authors

Correspondence to Chang Ming Song, Jian Lin Zhang or Yuan Yuan Wang.

Additional information

Supported by the NNSF of China (Grant Nos. 11671367 and 11801133), the Natural Science Foundation of Henan Province (Grant No. 152300410227) and the Key Research Projects of Henan Higher Education Institutions (Grant No. 18A110038)

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Song, C.M., Zhang, J.L. & Wang, Y.Y. Time-Periodic Solution to the Compressible Navier-Stokes/Allen-Cahn System. Acta. Math. Sin.-English Ser. 36, 419–442 (2020). https://doi.org/10.1007/s10114-020-9413-2

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  • DOI: https://doi.org/10.1007/s10114-020-9413-2

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