Inventiones mathematicae

, Volume 214, Issue 3, pp 1205–1266 | Cite as

Expanding large global solutions of the equations of compressible fluid mechanics

  • Mahir HadžićEmail author
  • Juhi Jang


Without any symmetry assumptions on the initial data we construct global-in-time unique solutions to the vacuum free boundary three-dimensional isentropic compressible Euler equations when the adiabatic exponent \(\gamma \) lies in the interval \((1,\frac{5}{3}]\). Our initial data lie sufficiently close to the expanding compactly supported affine motions recently constructed by Sideris and they satisfy the physical vacuum boundary condition.



The authors express their gratitude to P. Raphaël for fruitful discussions and for pointing out connections to the treatment of self-similar singular behavior for nonlinear Schrödinger equations. They also thank C. Dafermos for his feedback and pointing out important references. JJ is supported in part by NSF Grants DMS-1608492 and DMS-1608494 and a von Neumann fellowship of the Institute for Advanced Study through the NSF grant DMS-1128155. MH acknowledges the support of the EPSRC Grant EP/N016777/1.


  1. 1.
    Chen, G.-Q.: Remarks on R. J. DiPerna’s paper: convergence of the viscosity method for isentropic gas dynamics [Comm. Math. Phys. 91 (1983), no. 1, 1–30; MR0719807 (85i:35118)]. Proc. Am. Math. Soc. 125(10), 2981–2986 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68(7), 1157–1190 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids. EMS Monographs in Mathematics. EMS Publishing House, Zürich (2007)CrossRefGoogle Scholar
  4. 4.
    Christodoulou, D., Miao, S.: Compressible Flow and Euler’s Equations, Surveys in Modern Mathematics, vol. 9. International Press, Vienna (2014)zbMATHGoogle Scholar
  5. 5.
    Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum. Commun. Pure Appl. Math. 64(3), 328–366 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Coutand, D., Shkoller, S.: Well-posedness in smooth function spaces for the moving boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. 206(2), 515–616 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dacorogna, B., Moser, J.: On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(1), 1–26 (1990)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dafermos, C.: Hyperbolic Conservation Laws in Continuum Physics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 2nd edn. Springer, Berlin (2005)Google Scholar
  9. 9.
    DiPerna, R.J.: Convergence of the viscosity method for isentropic gas dynamics. Commun. Math. Phys. 91, 1–30 (1983)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Eggers, J., Fontelos, A.M.: The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1–R44 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Friedrich, H.: Sharp asymptotics for Einstein-\(\lambda \)-dust flows. Commun. Math. Phys. 350, 803–844 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Grassin, M.: Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J. 47, 1397–1432 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hadžić, M., Jang, J.: Nonlinear stability of expanding star solutions in the radially-symmetric mass-critical Euler-Poisson system. Commun. Pure Appl. Math. 71(5), 827–891 (2018)CrossRefGoogle Scholar
  14. 14.
    Hadžić, M., Speck, J.: The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant. J. Hyp. Differ. Equ. 12(1), 87–188 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Huang, H., Marcati, P., Pan, R.: Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 176, 1–24 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jang, J., Masmoudi, N.: Well-posedness for compressible Euler equations with physical vacuum singularity. Commun. Pure Appl. Math. 62, 1327–1385 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jang, J., Masmoudi, N.: Vacuum in gas and fluid dynamics. In: Proceedings of the IMA Summer School on Nonlinear Conservation Laws and Applications, pp. 315–329. Springer (2011)Google Scholar
  18. 18.
    Jang, J., Masmoudi, N.: Well and ill-posedness for compressible Euler equations with vacuum. J. Math. Phys. 53(11), 115625 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Jang, J., Masmoudi, N.: Well-posedness of compressible Euler equations in a physical vacuum. Commun. Pure Appl. Math. 68(1), 61–111 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kufner, A., Malgranda, L., Persson, L.-E.: The Hardy Inequality. Vydavatelský Servis, Plzen (2007)Google Scholar
  21. 21.
    Lindblad, H.: Well posedness for the motion of a compressible liquid with free surface boundary. Commun. Math. Phys. 260, 319–392 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lions, P.L., Perthame, B., Souganidis, P.E.: Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Commun. Pure Appl. Math. 49(6), 599–638 (1996)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liu, T.-P.: Compressible flow with damping and vacuum. Jpn. J. Appl. Math. 13, 25–32 (1996)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, T.-P., Smoller, J.: On the vacuum state for isentropic gas dynamics equations. Adv. Math. 1, 345–359 (1980)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Liu, T.-P., Yang, T.: Compressible Euler equations with vacuum. J. Differ. Equ. 140, 223–237 (1997)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Liu, T.-P., Yang, T.: Compressible flow with vacuum and physical singularity. Methods Appl. Anal. 7, 495–509 (2000)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Lübbe, C., Valiente-Kroon, J.A.: A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies. Ann. Phys. 328, 1–25 (2013)CrossRefGoogle Scholar
  28. 28.
    Luk, J., Speck, J.: Shock formation in solutions to the \(2D\) compressible Euler equations in the presence of non-zero vorticity. Invent. Math. (to appear). arXiv:1610.00737
  29. 29.
    Luo, T., Xin, Z., Zeng, H.: Well-posedness for the motion of physical vacuum of the three-dimensional compressible Euler equations with or without self-gravitation. Arch. Ration. Mech. Anal. 213(3), 763–831 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Luo, T., Zeng, H.: Global existence of smooth solutions and convergence to barenblatt solutions for the physical vacuum free boundary problem of compressible euler equations with damping. Commun. Pure Appl. Math. 69(7), 1354–1396 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Majda, A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Volume 53 of Applied Mathematical Sciences. Springer, New York (1984)CrossRefGoogle Scholar
  32. 32.
    Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Commun. Pure Appl. Math. 39(S, suppl.), S187–S220 (1986)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Makino, T., Ukai, S., Kawashima, S.: Sur la solution à support compact de l’équations d’Euler compressible. Jpn. J. Appl. Math. 3, 249–257 (1986)CrossRefGoogle Scholar
  34. 34.
    Merle, F., Raphaël, P., Szeftel, J.: Stable self similar blow up dynamics for \(L^2\)-supercritical NLS equations. Geom. Funct. Anal. 20(4), 1028–1071 (2010)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Oliynyk, T.: Future stability of the FLRW fluid solutions in the presence of a positive cosmological constant. Commun. Math. Phys. 346, 293–312 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Rodnianski, I., Speck, J.: The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant. J. Eur. Math. Soc. 15(6), 2369–2462 (2013)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Rozanova, O.: Solutions with Linear Profile of Velocity to the Euler Equations in Several Dimensions. Hyperbolic Problems: Theory, Numerics, Applications, pp. 861–870. Springer, Berlin (2003)zbMATHGoogle Scholar
  38. 38.
    Shkoller, S., Sideris, T.C.: Global existence of near-affine solutions to the compressible Euler equations. Preprint arXiv:1710.08368
  39. 39.
    Serre, D.: Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. l’Inst. Fourier 47, 139–153 (1997)CrossRefGoogle Scholar
  40. 40.
    Serre, D.: Expansion of a compressible gas in vacuum. Bull. Inst. Math. Acad. Sin. Taiwan 10, 695–716 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Speck, J.: The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant. Sel. Math. 18(3), 633–715 (2012)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Sideris, T.C.: Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. 101(4), 475–485 (1985)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sideris, T.C.: Spreading of the free boundary of an ideal fluid in a vacuum. J. Differ. Equ. 257(1), 1–14 (2014)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Sideris, T.C.: Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum. Arch. Ration. Mech. Anal. 225(1), 141–176 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College LondonStrand, LondonUK
  2. 2.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  3. 3.Korea Institute for Advanced StudySeoulKorea

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