Advertisement

Low Mach number limit of the three-dimensional full compressible Navier–Stokes–Korteweg equations

  • Kaijian Sha
  • Yeping LiEmail author
Article
  • 10 Downloads

Abstract

In this paper, we justify the low Mach number limit for the three-dimensional full compressible Navier–Stokes–Korteweg equations rigorously within the framework of smooth solution. Under the assumptions of small density and temperature perturbation, we show that for sufficiently small Mach number, the initial-value problem of the three-dimensional full compressible Navier–Stokes–Korteweg equations admits a unique smooth solution on the time interval where the smooth solution of the corresponding incompressible Navier–Stokes equations exists. Moreover, we obtain the convergence of smooth solutions for the full compressible Navier–Stokes–Korteweg equations toward those for the incompressible Navier–Stokes equations with a convergence rate.

Keywords

Full compressible Navier–Stokes–Korteweg equations Low Mach number limit Incompressible Navier–Stokes equations Error estimate 

Mathematics Subject Classification

35B35 35B40 76N15 

Notes

Acknowledgements

We are grateful to two anonymous referees for valuable comments which greatly improved our original manuscript. The research is supported in partial by the National Science Foundation of China (Grant No. 11671134).

References

  1. 1.
    Anderson, D.M., McFadden, G.B., Wheeler, G.B.: Diffuse-interface methods in fluid mech. Ann. Rev. Fluid Mech. 30, 139–165 (1998)zbMATHCrossRefGoogle Scholar
  2. 2.
    Alazard, T.: Low Mach number limit of the full Navier–Stokes equations. Arch. Ration. Mech. Anal. 180, 1–73 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alazard, T.: A minicourse on the low Mach number limit. Discrete Contin. Dyn. Syst. Ser. S 1, 365–404 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bresch, D., Desjardins, B.: On the construction of approximate solutions for the \(2D\) viscous shallow water model and for compressible Navier–Stokes models. J. Math. Pures Appl. (9) 86, 362–368 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bresch, D., Desjardins, B.: On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl. (9) 87, 57–90 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bresch, D., Desjardins, B., Lin, C.-K.: On some compressible fluid models: Korteweg, lubrication and shallow water systems. Commun. Partial Differ. Equ. 28, 843–868 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cheng, B.: Improved accuracy of incompressible approximation of compressible Euler equations. SIAM J. Math. Anal. 46, 3838–3864 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chen, Z.-Z., Chai, X.-J., Dong, B.-Q., Zhao, H.-J.: Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data. J. Differ. Equ. 259, 4376–4411 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Charve, F., Haspot, B.: Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system. SIMA J. Math. Anal. 45, 469–494 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1998)CrossRefGoogle Scholar
  11. 11.
    Cai, H., Tan, Z., Xu, Q.-J.: Time periodic solutions of the non-isentropic compressible fluid models of Korteweg type. Kinet. Relat. Model. 8, 29–51 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Chen, Z.-Z., Zhao, H.-J.: Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system. J. Math. Pures Appl. 101, 330–371 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Danchin, R.: Low Mach number limit for viscous compressible flows. Math. Model. Numer. Anal. 39, 459–475 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Danchin, R., Desjardins, B.: Existence of solutions for compressible fluid models of Korteweg type. Ann. Inst. Henri Poincaré Anal. Non linéaire 18, 97–133 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dunn, J.E., Serrin, J.: On the thermodynamics of interstitial working. Arch. Ration. Mech. Anal. 88, 95–133 (1985)zbMATHCrossRefGoogle Scholar
  16. 16.
    Germain, P., LeFloch, P.G.: Finite energy method for compressible fluids: the Navier–Stokes–Korteweg model. Commun. Pure Appl. Math. 69, 3–61 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Haspot, B.: Existence of strong solutions for nonisothermal Korteweg system. Annales Mathématiques Blaise Pascal 16, 431–481 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Haspot, B.: Existence of global strong solution for the compressible Navier–Stokes system and the Korteweg system in two-dimension. Methods Appl. Anal. 20, 141–164 (2013)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Haspot, B.: Existence of global weak solution for compressible fluid models of Korteweg type. J. Math. Fluid Mech. 13, 223–249 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hattori, H., Li, D.: The existence of global solutions to a fluid dynamic model for materials for Korteweg type. J. Partial Differ. Equ. 9, 323–342 (1996)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hattori, H., Li, D.: Golobal solutions of a high dimensional system for Korteweg materials. J. Math. Anal. Appl. 198, 84–97 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hou, X.-F., Peng, H.-Y., Zhu, C.-J.: Global well-posedness of the \(3D\) non-isothermal compressible fluid model of Korteweg type. Nonlinear Anal. Real World Appl. 43, 18–53 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hou, X.-F., Yao, L., Zhu, C.-J.: Vanishing capillarity limit of the compressible non-estropic Navier–Stokes–Korteweg system to the Navier–Stokes equations. J. Math. Anal. Appl. 448, 421–446 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Jiang, S., Ju, Q.-C., Li, F.-C.: Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations. Nonlinearity 25, 1351–1365 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Jiang, S., Ju, Q.-C., Li, F.-C., Xin, Z.-P.: Low Mach number limit for the full magnetohydrodynamic equations with general initial data. Adv. Math. 259, 384–420 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kato, T.: Nonstationary flow of viscous and ideal fluids in \({\mathbb{R}}^3\). J. Funct. Anal. 9, 296–305 (1972)zbMATHCrossRefGoogle Scholar
  27. 27.
    Korteweg, D.J.: Sur la forme que prennent les équations des mouvement des fluids si l’on tient comple des forces capillaries par des variations de densité. Arch. Neerl. Sci. Exactes Nat. Ser. II(6), 1–24 (1901)zbMATHGoogle Scholar
  28. 28.
    Kotschote, M.: Strong well-posedness for a Korteweg type for the dynamics of a compressible non-isothermal fluid. J. Math. Fluid Mech. 12, 473–483 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Kotschote, M.: Existence and time-asymptotics of global strong solutions to dynamic Korteweg models. Indiana Univ. Math. J. 63, 21–51 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Klainerman, S., Majda, A.: Singular limits of quasilinear hydrobolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)zbMATHCrossRefGoogle Scholar
  31. 31.
    Li, Y.-P., Liao, J.: Existence of strong solutions to the stationary compressible Navier–Stokes–Korteweg system with large external force. Nonlinear Anal. Real World Appl. 47, 204–223 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Lions, P.L., Masmoudi, N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Li, Y.-P., Yong, W.-A.: Zero Mach number limit of the compressible Navier–Stokes–Korteweg equations. Commun. Math. Sci. 14, 233–247 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Dafermos, C., Feireisl, E. (eds.) Handbook of Differential Equations. III. Elsevier/North-Holland, Amsterdam (2006)Google Scholar
  35. 35.
    Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)zbMATHCrossRefGoogle Scholar
  36. 36.
    McGrath, F.J.: Nonstationary plane flow of viscous and ideal fluds. Arch. Ration. Mech. Anal. 27, 229–348 (1968)CrossRefGoogle Scholar
  37. 37.
    Schochet, S.: The mathematical theory of low Mach number flows. Math. Model. Numer. Anal. 39, 441–458 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Van der Waals, J.D.: Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung. Z. Phys. Chem. 13, 657–725 (1894)Google Scholar
  39. 39.
    Yong, W.-A.: Basic aspects of hyperbolic relaxation systems. In: Freistuhler, H., Szepessy, A. (eds.) Advances in the Theory of Shock Waves. Progress in Nonlinear Differential Equations Applications, 47, pp. 259–305. Birkhauser Boston, Boston (2001)CrossRefGoogle Scholar
  40. 40.
    Yong, W.-A.: Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Differ. Equ. 155, 89–132 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Zhang, X., Tan, Z.: Decay estimates of the non-isentropic compressible fluid models of Korteweg type in \(\mathbb{R}^3\). Commun. Math. Sci. 12, 1437–1456 (2014)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsEast China University of Science and TechnologyShanghaiPeople’s Republic of China

Personalised recommendations