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Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations

Abstract

The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper. These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.

First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish. Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to the non-vacuum equilibrium state exponentially in time.

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References

  1. 1

    Balian R.: From microphysics to macrophysics. Springer, Berlin (1982)

    Google Scholar 

  2. 2

    Bresch D., Desjardins B.: Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238, 211–223 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  3. 3

    Bresch D., Desjardins B.: Some diffusive capillary models for Korteweg type. C. R. Mecanique 332(11), 881–886 (2004)

    Google 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. 86(4), 362–368 (2006)

    MATH  MathSciNet  Google Scholar 

  5. 5

    Bresch D., Desjardins B., Gérard-Varet D.: On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl. 87(2), 227–235 (2007)

    MATH  MathSciNet  Google Scholar 

  6. 6

    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. 87(1), 57–90 (2007)

    MATH  MathSciNet  Google Scholar 

  7. 7

    Bresch D., Desjardins B., Lin C.-K.: On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial Differ. Eqs. 28, 843–868 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  8. 8

    Bresch D., Desjardins B., Métivier G.: Recent Mathematical Results and Open Problems About Shallow Water Equations. Birkauser, Basel-Boston (2007)

    Google Scholar 

  9. 9

    Cho Y., Choe H.J., Kim H.: Unique solvability of the initial boundary value problems for compressible viscous fluids. J. Math. Pures Appl. 83(2), 243–275 (2004)

    MATH  MathSciNet  Google Scholar 

  10. 10

    Choe H.J., Kim H.: Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Eqs. 190(2), 504–523 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11

    Danchin R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141, 579–614 (2000)

    MATH  MathSciNet  Google Scholar 

  12. 12

    Fang D., Zhang T.: Compressible Navier-Stokes equations with vacuum state in one dimension. Comm. Pure Appl. Anal. 3, 675–694 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  13. 13

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

    MATH  Article  ADS  MathSciNet  Google Scholar 

  14. 14

    Gerbeau J.-F., Perthame B.: Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B1, 89–102 (2001)

    MathSciNet  Google Scholar 

  15. 15

    Guo, Z.-H., Jiang, S., Xie, F.: Global existence and asymptotic behavior of weak solutions to the 1D compressible Navier-Stokes equations with degenerate viscosity coefficient. Submitted for publication, 2007

  16. 16

    Guo Z.-H., Jiu Q.-S., Xin Z.: Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J. Math. Anal. 39(5), 1402–1427 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. 17

    Guo, Z.-H., Zhu, C.-J.: Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Submitted for publication, 2007

  18. 18

    Hoff D.: Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Amer. Math. Soc. 303(1), 169–181 (1987)

    MATH  Article  MathSciNet  Google Scholar 

  19. 19

    Hoff D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rat. Mech. Anal. 132, 1–14 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  20. 20

    Hoff D., Serre D.: The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math. 51(4), 887–898 (1991)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  21. 21

    Hoff D., Smoller J.: Non-formation of vacuum states for compressible Navier-Stokes equations. Commun. Math. Phys. 216(2), 255–276 (2001)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  22. 22

    Huang F., Li J., Xin Z.: Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data. J. Math. Pures Appl. 86(6), 471–491 (2006)

    MATH  MathSciNet  Google Scholar 

  23. 23

    Jiang S., Xin Z., Zhang P.: Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods and Applications of Analysis 12, 239–251 (2005)

    MathSciNet  Google Scholar 

  24. 24

    Kazhikhov A.V., Shelukhin V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41(2), 273–282 (1977)

    MATH  Article  MathSciNet  Google Scholar 

  25. 25

    Ladyzenskaja, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. Providence, R.I.: American Mathematical Society, 1968

  26. 26

    Li, H.-L., Li, J., Xin, Z.: Vanishing vacuum states and blow-up of global weak solutions for full Navier-Stokes equations. In preparation, 2006

  27. 27

    Li, J.: Qualitative behavior of solutions to the compressible Navier-Stokes equations and its variants. PhD Thesis, Chinese University of Hong Kong, 2004

  28. 28

    Li J., Xin Z.: Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows. J. Differ. Eqs. 221, 275–308 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  29. 29

    Lions P.L.: Existence globale de solutions pour les equations de Navier-Stokes compressibles isentropiques. C. R. Acad. Sci. Paris, Sér I Math. 316, 1335–1340 (1993)

    MATH  Google Scholar 

  30. 30

    Lions P.L.: Limites incompressible et acoustique pour des fluides visqueux, compressibles et isentropiques. C. R. Acad. Sci. Paris Sér. I Math. 317, 1197–1202 (1993)

    MATH  Google Scholar 

  31. 31

    Lions P.-L.: Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  32. 32

    Liu T.-P., Xin Z., Yang T.: Vacuum states for compressible flow. Discrete Contin. Dynam. Systems 4, 1–32 (1998)

    MATH  MathSciNet  Google Scholar 

  33. 33

    Luo T., Xin Z., Yang T.: Interface behavior of compressible Navier-Stokes equations with vacuum. SIAM J. Math. Anal. 31, 1175–1191 (2000)

    MATH  Article  MathSciNet  Google Scholar 

  34. 34

    Marche F.: Derivation of a new two-dimensional shallow water model with varying topography, bottom friction and capillary effects. Eur. J. Mech. B Fluids 26(1), 49–63 (2007)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  35. 35

    Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci. 55(9), 337–342 (1979)

    MATH  MathSciNet  Google Scholar 

  36. 36

    Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)

    MATH  MathSciNet  Google Scholar 

  37. 37

    Matsumura A., Nishida T.: The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids. Commun. Math. Phys. 89, 445–464 (1983)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  38. 38

    Mellet A., Vasseur A.: On the barotropic compressible Navier-Stokes equations. Commun. Partial Differ. Eqs. 32(1–3), 431–452 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  39. 39

    Mellet A., Vasseur A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations. SIAM J. Math. Anal. 39(4), 1344–1365 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. 40

    Nishida, T.: Motion of compressible fluids. In: “Patterns and waves. Qualitative analysis of nonlinear differential equations”, Ed. Nishida, Masayasu Mimura and Hiroshi Fujii, Tokyo: Kinokuniya Company Ltd., 1986, pp. 89–19

  41. 41

    Okada M.: Free boundary problem for the equation of one-dimensional motion of viscous gas. Japan J. Appl. Math. 6, 161–177 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  42. 42

    Okada M., Makino T.: Free boundary problem for the equation of spherically symmetric motion of viscous gas. Japan J. Appl. Math. 10, 219–235 (1993)

    MATH  MathSciNet  Google Scholar 

  43. 43

    Okada M., Matsusu-Necasova S., Makino T.: Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity. Ann. Univ. Ferrara Sez. VII (N.S.) 48, 1–20 (2002)

    MATH  MathSciNet  Google Scholar 

  44. 44

    Straškraba I., Zlotnik A.: Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity. Z. Angew. Math. Phys. 54(4), 593–607 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. 45

    Salvi R., Straškraba I.: Global existence for viscous compressible fluids and their behavior as t→ ∞. J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 40, 17–51 (1993)

    MATH  Google Scholar 

  46. 46

    Serre D.: Solutions faibles globales des quations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303(13), 639–642 (1986)

    MATH  MathSciNet  Google Scholar 

  47. 47

    Serre D.: On the one-dimensional equation of a viscous, compressible, heat-conducting fluid. C. R. Acad. Sci. Paris Sér. I Math. 303(14), 703–706 (1986)

    MATH  MathSciNet  Google Scholar 

  48. 48

    Solonnikov V.A.: On solvability of an initial boundary value problem for the equations of motion of viscous compressible fluid. Zap. Nauchn. Sem. LOMI 56, 128–142 (1976)

    MathSciNet  MATH  Google Scholar 

  49. 49

    Vaigant V.A., Kazhikhov A.V.: On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. (Russian). Sibirsk. Mat. Zh. 36(6), 1283–1316 (1995)

    MathSciNet  Google Scholar 

  50. 50

    Valli A., Zajaczkowski W.M.: Navier-Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103(2), 259–296 (1986)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  51. 51

    Vong S.-W., Yang T., Zhu C.: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. II. J. Differ. Eqs. 192(2), 475–501 (2003)

    MATH  Article  MathSciNet  Google Scholar 

  52. 52

    Xin Z.: Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. 51, 229–240 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  53. 53

    Xin, Z.: On the behavior of solutions to the compressible Navier-Stokes equations. AMS/IP Stud. Adv. Math. 20, Providence, RI: Amen. Math. Soc., 2001, pp. 159–170

  54. 54

    Xin Z., Yuan H.: Vacuum state for spherically symmetric solutions of the compressible Navier-Stokes equations. J Hyperbolic Differ. Eqs. 3, 403–442 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  55. 55

    Yang T., Yao Z., Zhu C.: Compressible Navier-Stokes equations with density-dependent viscosity and vacuum. Comm. Partial Differ Eqs. 26(5–6), 965–981 (2001)

    MATH  Article  MathSciNet  Google Scholar 

  56. 56

    Yang T., Zhao H.: A vacuum problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. J. Differ. Eqs. 184, 163–184 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  57. 57

    Yang T., Zhu C.: Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum. Commun. Math. Phys. 230, 329–363 (2002)

    MATH  Article  ADS  MathSciNet  Google Scholar 

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Correspondence to Hai-Liang Li.

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Communicated by P. Constantin

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Li, HL., Li, J. & Xin, Z. Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations. Commun. Math. Phys. 281, 401 (2008). https://doi.org/10.1007/s00220-008-0495-4

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Keywords

  • Weak Solution
  • Vacuum State
  • Dirichlet Boundary Condition
  • Global Existence
  • Global Weak Solution