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Global Spherically Symmetric Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System

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Abstract

In this paper, we consider the global spherically symmetric solutions for the initial boundary value problem of a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of two-phase viscous compressible fluids. We prove the existence and uniqueness of global classical solution, weak solution and strong solution under the assumption of spherically symmetry condition for initial data ρ0 without vacuum state.

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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 compressible viscous fluids. Indiana Univ. Math. J. 57(2), 659–698 (2008)

    Article  MathSciNet  Google Scholar 

  3. Anderson, D. M., McFadden, G. B., Wheeler, A. A.: Diffuse-interface method in fluid mechanics. Annu. Rev. Fluid Mech., 30, 139–165 (1998)

    Article  MathSciNet  Google Scholar 

  4. Babak, S., Stefan, T., Matthias, M., et al.: Isogeometric analysis of the Navier–Stokes/Cahn–Hilliard. J. Comput. Phys., 348, 171–194 (2017)

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  8. Chen, Y., He, Q., Huang, B., et al.: Global strong solution to a thermodynamic compressible diffuse interface model with temperature dependent heat-conductivity in 1-D. Math. Methods Appl. Sci. 44(17), 12945–12962 (2021)

    Article  MathSciNet  Google Scholar 

  9. Chen, Y., He, Q., Huang, B., et al.: The Cauchy problem for non-isentropic compressible Navier–Stokes/Allen–Cahn system with degenerate heat-conductivity. arXiv:2005.11205 (2020)

  10. Chen, M., Guo, X.: 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 

  11. 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 

  12. Chen, S., Zhu, C.: Blow-up criterion and the global existence of strong/classical solutions to Navier–Stokes/Allen–Cahn system. Z. Angew. Math. Phys., 72(1), Article number: 14 (2021)

  13. Ding, S., Li, Y., Tang, Y.: Strong solutions to 1D compressible Navier–Stokes/Allen–Cahn system with free boundary. Math. Methods Appl. Sci. 42(14), 4780–4794 (2019)

    Article  MathSciNet  Google Scholar 

  14. Deugoue, G., Tachim, M. T.: Convergence of the solution of the stochastic 3D globally modified Navier–Stokes/Cahn–Hilliard equations. J. Differential Equations, 265, 545–592 (2018)

    Article  MathSciNet  Google Scholar 

  15. Ding, S., Li, Y., Luo, W.: Global solutions for a coupled compressible Navier–Stokes/Allen–Cahn system in 1D. J. Math. Fluid Mech., 15, 335–360 (2013)

    Article  MathSciNet  Google Scholar 

  16. Ding, S., Lin, J., Wang, C., et al.: Compressible hydrodynamic flow liquid crystals in 1D. Discrete Contin. Dyn. Syst. 32(2), 539–563 (2012)

    Article  MathSciNet  Google Scholar 

  17. Feireisl, E., Petzeltova, H., Rocca, E., et al.: Analysis of a phase-field model for two-phase compressible fluids. Math. Methods Appl. Sci., 31, 1972–1995 (2008)

    MathSciNet  Google Scholar 

  18. Feng, J. J., Liu, C., Shen, J., et al.: An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges. Modeling of Soft Matter IMA Vol Math. Appl., 141, 1–26 (2005)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Gal, C. G., Grasselli, M.: Trajectory attractors for binary fluid mixtures in 3D. Chin. Ann. Math. Ser. B 31(5), 655–678 (2010)

    Article  MathSciNet  Google Scholar 

  21. 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. Phys. D, 240, 629–635 (2011)

    Article  MathSciNet  Google Scholar 

  22. Giorgini, A., Miranville, A., Temam, R.: Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system. SIAM J. Math. Anal., 51, 2535–2574 (2019)

    Article  MathSciNet  Google Scholar 

  23. Kotschote, M.: Strong solutions of the Navier–Stokes equations for a compressible fluid of Allen–Cahn type. Arch. Ration. Mech. Anal., 206, 489–514 (2012)

    Article  MathSciNet  Google Scholar 

  24. 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 

  25. Li, Y., Huang, M.: Strong solutions for an incompressible Navier–Stokes/Allen–Cahn system with different densities. Z. Angew. Math. Phys., 69, Article number: 68 (2018)

  26. 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 

  27. Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 454, 2617–2654 (1998)

    Article  MathSciNet  Google Scholar 

  28. Simon, J.: Nonhomogeneous viscous incompressible fluids: Existence of velocity, density and pressure. SIAM J. Math. Anal., 21, 1093–1117 (1990)

    Article  MathSciNet  Google Scholar 

  29. Song, C., Wang, Y.: Spherically symmetric solutions for a coupled compressible Navier–Stokes/Allen–Cahn system. J. Partial Differential Equations 31(4), 374–384 (2018)

    Article  MathSciNet  Google Scholar 

  30. Song, C., Zhang, J., Wang, Y.: Time-periodic solution to the compressible Navier–Stokes/Allen–Cahn system. Acta Math. Sinica, Engl. Ser. 36(4), 419–442 (2020)

    Article  MathSciNet  Google Scholar 

  31. Tachim, M. 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 

  32. 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 

  33. Witterstein, G., Hoffmann, K. H.: 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 

  34. Xu, X., Zhao, L., Liu, C.: Axisymmetric solutions to coupled Navier–Stokes/Allen–Cahn equations. SIAM J. Math. Anal. 41(6), 2246–2282 (2010)

    Article  MathSciNet  Google Scholar 

  35. Yan, Y., Ding, S., Li, Y.: Strong solutions for 1D compressible Navier–Stokes/Allen–Cahn system with phase variable dependent viscosity. arXiv:2109.14529 (2021)

  36. Yang, X., Feng, 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 

  37. Zhang, J.: A regularity criterion for the 3D incompressible density-dependent Navier–Stokes-Allen–Cahn equations. J. Partial Differential Equations 29(2), 117–124 (2016)

    MathSciNet  Google Scholar 

  38. Zhang, J.: Regularity of solutions to 1D compressible Navier–Stokes-Allen–Cahn system. Appl. Anal. 100(9), 1827–1842 (2021)

    Article  MathSciNet  Google Scholar 

  39. 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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Acknowledgements

We thank the referees for their time and comments.

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Authors and Affiliations

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

    Chang Ming Song, Jian Lin Zhang & Yuan Yuan Wang

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  1. Chang Ming Song
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  2. Jian Lin Zhang
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  3. Yuan Yuan Wang
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Correspondence to Chang Ming Song.

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Conflict of Interest The authors declare no conflict of interest.

Additional information

Supported by the NNSF of China (Grant Nos. 12171438, 11801133), the Natural Science Foundation of Henan Province (Grant No. 152300410227) and the grant from the Special Project of Basic Scientific Research Business Expenses of Zhongyuan University of Technology (Grant No. K2020TD004)

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Song, C.M., Zhang, J.L. & Wang, Y.Y. Global Spherically Symmetric Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-3399-0

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  • DOI: https://doi.org/10.1007/s10114-024-3399-0

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