Global Existence of Regular Solutions with Large Oscillations and Vacuum

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Abstract

The global existence of smooth solutions to the compressible Navier-Stokes equations is investigated. In particular, results are reviewed concerning the global existence and uniqueness of classical solutions to the Cauchy problem for the barotropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. The initial density is allowed to vanish and the spatial measure of the vacuum set can be arbitrarily large, in particular, the initial density can even have compact support. These results generalize previous ones on classical solutions for initial densities being strictly away from vacuum, and are the first for global classical solutions that may have large oscillations and can contain vacuum states.

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References

  1. 1.
    J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)Google Scholar
  2. 2.
    J. Bergh, J. Lofstrom, Interpolation Spaces. An Introduction (Springer, Berlin/Heidelberg/New York, 1976)Google Scholar
  3. 3.
    Y. Cho, H.J. Choe, H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluid. J. Math. Pures Appl. 83, 243–275 (2004)Google Scholar
  4. 4.
    Y. Cho, B.J. Jin, Blow-up of viscous heat-conducting compressible flows. J. Math. Anal. Appl. 320(2), 819–826 (2006)Google Scholar
  5. 5.
    Y. Cho, H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscr. Math. 120, 91–129 (2006)Google Scholar
  6. 6.
    H.J. Choe, H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids. J. Differ. Equ. 190, 504–523 (2003)Google Scholar
  7. 7.
    R. Danchin, Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141, 579–614 (2000)Google Scholar
  8. 8.
    Q. Duan, Global well-posedness of classical solutions to the compressible Navier-Stokes equations in a half-space. J. Differ. Equs. 253(1), 167–202 (2012)Google Scholar
  9. 9.
    E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech. 3(4), 358–392 (2001)Google Scholar
  10. 10.
    H. Fujita, T. Kato, On the Navier-Stokes initial value problem I. Arch. Rational Mech. Anal. 16, 269–315 (1964)Google Scholar
  11. 11.
    D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data. Trans. Amer. Math. Soc. 303(1), 169–181 (1987)Google Scholar
  12. 12.
    D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)Google Scholar
  13. 13.
    D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rational Mech. Anal. 132, 1–14 (1995)Google Scholar
  14. 14.
    D. Hoff, Compressible flow in a half-space with Navier boundary conditions. J. Math. Fluid Mech. 7(3), 315–338 (2005)Google Scholar
  15. 15.
    D. Hoff, Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 55(11), 1365–1407 (2002)Google Scholar
  16. 16.
    D. Hoff, M.M. Santos, Lagrangean structure and propagation of singularities in multidimensional compressible flow. Arch. Rational Mech. Anal. 188(3), 509–543 (2008)Google Scholar
  17. 17.
    D. Hoff, E. Tsyganov, Time analyticity and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional compressible flow. J. Differ. Equ. 245(10), 3068–3094 (2008)Google Scholar
  18. 18.
    X. Huang, J. Li, A. Matsumura, On the strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum (2016, submitted)Google Scholar
  19. 19.
    X.D. Huang, J. Li, Z.P. Xin, Serrin type criterion for the three-dimensional compressible flows. SIAM J. Math. Anal. 43(4), 1872–1886 (2011)Google Scholar
  20. 20.
    X.D. Huang, J. Li, Z.P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)Google Scholar
  21. 21.
    X.D. Huang, Z.P. Xin, A blow-up criterion for classical solutions to the compressible Navier-Stokes equations. Sci. China 53(3), 671–686 (2010)Google Scholar
  22. 22.
    D. Jesslé, B.J. Jin, A. Novotný, Navier-Stokes-Fourier system on unbounded domains: weak solutions, relative entropies, weak-strong uniqueness. SIAM J. Math. Anal. 45(3), 1907–1951 (2013)Google Scholar
  23. 23.
    A.V. Kazhikhov, V.V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh. 41, 282–291 (1977)Google Scholar
  24. 24.
    T. Kato, Strong L p-solutions of the Navier-Stokes equation in R m, with applications to weak solutions. Math. Z. 187(4), 471–480 (1984)Google Scholar
  25. 25.
    H. Koch, D. Tataru, Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001)Google Scholar
  26. 26.
    O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and quasilinear equations of parabolic type (American Mathematical Society, Providence, 1968)Google Scholar
  27. 27.
    J. Li, Z. Liang, On local classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum. J. Math. Pures Appl. (9) 102(4), 640–671 (2014)Google Scholar
  28. 28.
    J. Li, Z. Xin, Some uniform estimates and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows. J. Differ. Equ. 221(2), 275–308 (2006)Google Scholar
  29. 29.
    J. Li, Z. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with Vacuum (2013). http://arxiv.org/abs/1310.1673
  30. 30.
    P.L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models (Oxford University Press, New York, 1998)Google Scholar
  31. 31.
    Z. Luo, Global existence of classical solutions to two-dimensional Navier-Stokes equations with Cauchy data containing vacuum. Math. Methods Appl. Sci. 37(9), 1333–1352 (2014)MathSciNetMATHGoogle Scholar
  32. 32.
    A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    J. Nash, Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France. 90, 487–497 (1962)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow (Oxford University Press, Oxford, 2004)MATHGoogle Scholar
  35. 35.
    O. Rozanova, Blow up of smooth solutions to the compressible Navier-Stokes equations with the data highly decreasing at infinity. J. Differ. Equ. 245, 1762–1774 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    R. Salvi, I. Straškraba, Global existence for viscous compressible fluids and their behavior as t →∞. J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 40, 17–51 (1993)Google Scholar
  37. 37.
    D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible. C. R. Acad. Sci. Paris Sér. I Math. 303, 639–642 (1986)MathSciNetMATHGoogle Scholar
  38. 38.
    D. Serre, Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math., 303, 703–706 (1986)MathSciNetMATHGoogle Scholar
  39. 39.
    J. Serrin, On the uniqueness of compressible fluid motion. Arch. Ration. Mech. Anal. 3, 271–288 (1959)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Z.P. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Z.P. Xin, W. Yan, On blowup of classical solutions to the compressible Navier-Stokes equations. Commun. Math. Phys. 321(2), 529–541 (2013)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    T. Zhang, Global solutions of compressible Navier-Stokes equations with a density-dependent viscosity coefficient. J. Math. Phys. 52, 043510 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    A.A. Zlotnik, Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differ. Equ. 36, 701–716 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Institute of Applied Mathematics, AMSS & Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingP. R. China
  2. 2.The Institute of Mathematical SciencesThe Chinese University of Hong KongHong KongChina

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