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Incompressible Limit for the Compressible Flow of Liquid Crystals

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Abstract

The connection between the compressible flow of liquid crystals with low Mach number and the incompressible flow of liquid crystals is studied in a bounded domain. In particular, the convergence of weak solutions of the compressible flow of liquid crystals to the weak solutions of the incompressible flow of liquid crystals is proved when the Mach number approaches zero; that is, the incompressible limit is justified for weak solutions in a bounded domain.

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References

  1. Alazard T.: Low Mach number limit of the full Navier–Stokes equations. Arch. Ration. Mech. Anal. 180(1), 1–73 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bresch D., Desjardins B., Grenier E., Lin C.K.: Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case. Stud. Appl. Math. 109(2), 125–149 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chandrasekhar S.: Liquid crystals. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  4. Danchin R.: Zero Mach number limit for compressible flows with periodic boundary conditions. Am. J. Math. 124(6), 1153–1219 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Desjardins B., Grenier E.: Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A. Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Desjardins B., Grenier E., Lions P. L., Masmoudi N.: Incompressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. (9) 78(5), 461–471 (1999)

    Article  MathSciNet  Google Scholar 

  7. Donatelli D., Marcati P.: A dispersive approach to the artificial compressibility approximations of the Navier–Stokes equations in 3D. J. Hyperb. Differ. Equ. 3(3), 575–588 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Donatelli D., Marcati P.: A quasineutral type limit for the Navier–Stokes–Poisson system with large data. Nonlinearity 21(1), 135–148 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. DE Gennes P.G.: The physics of liquid crystals. Oxford University Press, Oxford (1974)

    Google Scholar 

  10. Donatelli D., Trivisa K.: From the dynamics of gaseous stars to the incompressible Euler equations. J. Differ. Equ. 245(5), 1356–1385 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Ebin D.G.: The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math. (2) 105(1), 141–200 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ericksen K.L.: Hydrostatic theory of liquid crystal. Arch. Ration. Mech. Anal. 9, 371–378 (1962)

    MATH  MathSciNet  Google Scholar 

  13. Feireisl E., Novotný A.: The low Mach number limit for the full Navier–Stokes–Fourier system. Arch. Ration. Mech. Anal. 186(1), 77–107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feireisl E., Novotný A., Petzeltov H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Feireisl E., Novotný A., Petzeltov H.: On the incompressible limit for the Navier–Stokes–Fourier system in domains with vary bottoms. Math. Models Appl. Sci. 18(2), 291–324 (2008)

    Article  MATH  Google Scholar 

  16. Galdi G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, vol. 1. Linearized steady problems. Springer, New York (1994)

    Google Scholar 

  17. Grenier E.: Oscillatory pertubations of the Navier–Stokes equations. J. Math. Pures Appl. 76(9), 477–498 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hardt R., Kinderlehrer D.: Mathematical questions of liquid crystal theory. The IMA volumes in mathematics and its applications 5. Springer, New York (1987)

    Google Scholar 

  19. Hardt R., Kinderlehrer D., Lin F.H.: Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105(4), 547–570 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Hagstrom T., Lorenz J.: All-time existence of classical solutions for slightly compressible flows. SIAM J. Math. Anal. 29(3), 652–672 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hoff D.: The zero-Mach limit of compressible flows. Comm. Math. Phys. 192(3), 543–554 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  22. Hoff D.: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Comm. Pure Appl. Math. 55(11), 1365–1407 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hu X., Wang D.: Low Mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J. Math. Anal. 41(3), 1272–1294 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jiang S., Ju Q., Li F.: Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun. Math. Phys. 297(2), 371–400 (2010)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34(4), 481–524 (1981)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Klainerman S., Majda A.: Compressible and incompressible fluids. Comm. Pure Appl. Math. 35(5), 629–653 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Leslie F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28(4), 265–283 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lin C.K.: On the incompressible limit of the compressible Navier–Stokes equations. Comm. Partial Differ. Equ. 20(3–4), 677–707 (1995)

    Article  MATH  Google Scholar 

  29. Lin F.-H.: Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Comm. Pure Appl. Math. 42(6), 789–814 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lin F.-H., Liu C.: Nonparabolic dissipative systems modeling the flow of liquid crystals. Comm. Pure Appl. Math. 48(5), 501–537 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lin F.-H., Liu C.: Partial regularity of the dynamic system modeling the flow of liquid crystals. Discret. Contin. Dynam. Syst. 2(1), 1–22 (1996)

    MATH  Google Scholar 

  32. Liu X.-G., Qing J.: Globally weak solutions to the flow of compressible liquid crystals system. Discret. Contin. Dyn. Syst. 33(2), 757–788 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lions, P.-L.: Mathematical topics in fluid mechanics. vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1998)

  34. Lions J.-L.: Quelques méthodes de résolution des problèms aux limites nonlinéaires. Gauthier-Villars, Paris (1960)

    Google Scholar 

  35. Lions P.-L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures. Appl. (9) 77(6), 585–627 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. Masmoudi N.: Incompressible, inviscid limit of the compressible Navier–Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 199–224 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Métivier G., Schochet S.: The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158(1), 61–90 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Métivier G., Schochet S.: Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differ. Equ. 187(1), 106–183 (2003)

    Article  MATH  ADS  Google Scholar 

  39. Schochet S.: The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit. Comm. Math. Phys. 104(1), 49–75 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  40. Schochet S.: Fast singular limits of hyperbolic PDEs. J. Differ. Equ. 114(2), 476–512 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Schochet S.: The mathematical theory of low Mach number flows. Math. Model. Numer. Anal. 39(3), 441–458 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang S., Jiang J.: The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations. Comm. Partial Differ. Equ. 31(4-6), 571–591 (2006)

    Article  MATH  Google Scholar 

  43. Wang D., Yu C.: Global weak solution and large-time behavior for the compressible flow of liquid crystals. Arch. Ration. Mech. Anal. 204(3), 881–915 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Author notes

  1. Cheng Yu

    Present address: Department of Mathematics, University of Texas, Austin, TX, 78712, USA

Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Dehua Wang & Cheng Yu

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  1. Dehua Wang
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  2. Cheng Yu
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Correspondence to Dehua Wang.

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Communicated by G.-Q. Chen

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Wang, D., Yu, C. Incompressible Limit for the Compressible Flow of Liquid Crystals. J. Math. Fluid Mech. 16, 771–786 (2014). https://doi.org/10.1007/s00021-014-0185-2

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