Skip to main content
SpringerLink
Log in

Stability of the Phase Separation State for Compressible Navier-Stokes/Allen-Cahn System

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/Allen-Cahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time t → +∞.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Abels, H., Feireisl, E. On a diffuse interface model for a two-phase flow of compressible viscous fluids. Indiana Univ. Math J., 57(2): 569–578 (2008)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Chen, Y., He, Q., Huang, B., Shi, X. Global existence of Cauchy problem for non-isentropic Navier-Stokes/Allen-Cahn system in 1-D. arXiv:2005.11205v2 [math.AP]

  5. Chen, Y., He, Q., Mei, M., Shi, X. Asymptotic stability of solutions for 1-D compressible Navier-Stokes-Cahn-Hilliard system. J. Math. Anal. Appl., 467: 185–206 (2018)

    Article  MathSciNet  Google Scholar 

  6. Chen, S., Wen, H., Zhu, C. Global existence of weak solution to compressible Navier-Stokes/Allen-Cahn system in three dimensions. J. Math. Anal. Appl., 477: 1265–1295 (2019)

    Article  MathSciNet  Google Scholar 

  7. 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 

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

    Article  MathSciNet  Google Scholar 

  9. Favre, G., Schimperna, G. On a Navier-Stokes/Allen-Cahn model with inertial effects. J. Math. Anal. Appl., 475: 811–838 (2019)

    Article  MathSciNet  Google Scholar 

  10. Feireisl, E., Petzeltová, H., Rocca, E., Schimperna, G. Analysis of a phase-field model for two-phase compressible fluids. Math. Models Meth. Appl. Sci., 20(7): 1129–1160 (2010)

    Article  MathSciNet  Google Scholar 

  11. Freistühler, H. Phase transitions and traveling waves in compressible fluids. Arch. Rational Mech. Anal., 211: 189–204 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gao, J., Tao, Q., Yao, Z.-A. Long-time behavior of solution for the compressible nematic liquid crystal flows in ℝ3. J. Differential Equations, 261: 2334–2383 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Gal, C.G., Grasselli, M. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete Contin Dynam Systems Ser. A, 28: 1–39 (2010)

    Article  MathSciNet  Google Scholar 

  15. Guo, Y., Wang, Y.J. Decay of dissipative equations and negative Sobolev spaces. Commun. P.D.E., 37: 2165–2208 (2012)

    Article  MathSciNet  Google Scholar 

  16. Heida, M., Malek, J., Rajagopal, K.R. On the development and generalizations of Allen-Cahn and Stefan equations within a thermodynamic framework. Z. Angew. Math. Phys., 63: 759–776 (2012)

    Article  MathSciNet  Google Scholar 

  17. Jiang, J., Li, Y., Liu, C. Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete Contin Dynam Systems, 37: 3243–3284 (2017)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Kotschote, M. Spectral analysis for travelling waves in compressible two-phase fluids of Navier-Stokes/Allen-Cahn type. J. Evol. Equ., 17: 359–385 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Li, Y., Huang, M. Strong solutions for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Z. Angew. Math. Phys., 69(68): 18 pp. (2018)

  22. Li, Y., Ding, S., Huang, M. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete Contin Dynam Systems Ser B., 21: 1507–1523 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Luo, T., Yin, H., Zhu, C. Stability of the rarefaction wave for a coupled compressible Navier-Stokes/Allen-Cahn system. Math. Methods Appl. Sci., 41(12): 4724–4736 (2018)

    Article  MathSciNet  Google Scholar 

  25. Medjo, T.T. A non-autonomous two-phase flow model with oscillating external force and its global attractor. Nonlinear Analysis, 75: 226–243 (2012)

    Article  MathSciNet  Google Scholar 

  26. Nirenberg, L. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, 13: 115–162 (1959)

    MathSciNet  Google Scholar 

  27. Poláčik, P. Symmetry Properties of Positive Solutions of Parabolic Equations on RN: I. Asymptotic Symmetry for the Cauchy Problem. Communications in P.D.E., 30: 1567–1593 (2005)

    Article  Google Scholar 

  28. Pu, X., Guo, B. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinet. Relat. Models., 9(1): 165–191 (2016)

    Article  MathSciNet  Google Scholar 

  29. Schonbek, M.E. Large time behaviour of solutions to the Navier-Stokes equations in Hn spaces. Comm. Partial Differential Equations, 20: 103–117 (1995)

    Article  MathSciNet  Google Scholar 

  30. Stein, E.M. Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, 1970

    Google Scholar 

  31. Tan, Z., Zhang, R. Optimal decay rates of the compressible fluid models of Korteweg type. Z. Angew. Math. Phys., 65: 279–300 (2014)

    Article  MathSciNet  Google Scholar 

  32. Wang, Y. Decay of the Navier-Stokes-Poisson equations. J. Differential Equations, 253: 273–297 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. Yin, H., Zhu, C. Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier-Stokes/Allen-Cahn system. J. Differential Equations, 266: 7291–7326 (2019)

    Article  MathSciNet  Google Scholar 

  35. 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 

  36. Zhao, X. Global well-posedness and decay estimates for three-dimensional compressible Navier-Stokes-Allen-Cahn system. Proc. Roy. Soc. Edinb. A: Mathematics, 152(5): 1291–1322 (2022)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their careful comments and suggestions leading to improvements in the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, 100029, China

    Ya-zhou Chen & Xiao-ding Shi

  2. Institute of Mathematics, State Academy of Sciences, Pyongyang, Korea

    Hakho Hong

Authors
  1. Ya-zhou Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hakho Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-ding Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-ding Shi.

Ethics declarations

The authors declare no conflict of interest.

Additional information

This paper is supported by the National Natural Science Foundation of China (Nos. 12171024, 11901025, 11971217, 11971020); Academic and Technical Leaders Training Plan of Jiangxi Province (No. 20212BCJ23027).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Yz., Hong, H. & Shi, Xd. Stability of the Phase Separation State for Compressible Navier-Stokes/Allen-Cahn System. Acta Math. Appl. Sin. Engl. Ser. 40, 45–74 (2024). https://doi.org/10.1007/s10255-023-1070-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-023-1070-7

Keywords

2000 MR Subject Classification

Search

Navigation