Archive for Rational Mechanics and Analysis

, Volume 225, Issue 1, pp 141–176 | Cite as

Global Existence and Asymptotic Behavior of Affine Motion of 3D Ideal Fluids Surrounded by Vacuum



The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in \({{\rm GL}^+(3, \mathbb{R})}\). The evolution of the fluid domain is described by a family of ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank r = 1, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along 3−r of its principal axes. In the compressible case, the asymptotic limit has rank r = 3, and asymptotic completeness holds, when the adiabatic index \({\gamma}\) satisfies \({4/3 < \gamma < 2}\). The number of possible degeneracies, 3−r, increases with the value of the adiabatic index \({\gamma}\). In the incompressible case, affine motion reduces to geodesic flow in \({{\rm SL}(3, \mathbb{R})}\) with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16(fasc. 1), 319–361 (1966)Google Scholar
  2. 2.
    Chemin J.Y.: Dynamique des gaz à masse totale finie. Asymptot. Anal. 3(3), 215–220 (1990)MathSciNetMATHGoogle Scholar
  3. 3.
    Christodoulou, D.: The formation of black holes in general relativity. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2009). doi:10.4171/068
  4. 4.
    Christodoulou, D., Lindblad, H.: On the motion of the free surface of a liquid. Comm. Pure Appl. Math. 53(12), 1536–1602 (2000). doi:10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.3.CO;2-H
  5. 5.
    Coutand D., Lindblad H., Shkoller S.: A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum. Comm. Math. Phys. 296(2), 559–587 (2010). doi:10.1007/s00220-010-1028-5 ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Coutand D., Shkoller S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007). doi:10.1090/S0894-0347-07-00556-5 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Coutand D., Shkoller S.: A simple proof of well-posedness for the free-surface incompressible Euler equations. Discret. Contin. Dyn. Syst. Ser. S 3(3), 429–449 (2010). doi:10.3934/dcdss.2010.3.429 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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). doi:10.1007/s00205-012-0536-1 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Grassin M.: Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J. 47(4), 1397–1432 (1998). doi:10.1512/iumj.1998.47.1608 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hadžić, M., Jang, J.: Expanding Large Global Solutions of the Equations of Compressible Fluid Mechanics. arXiv:1610.01666 (2016).
  11. 11.
    Jang, J., Masmoudi, N.: Well and ill-posedness for compressible Euler equations with vacuum. J. Math. Phys. 53(11), 115,625 (2012). doi:10.1063/1.4767369
  12. 12.
    Jang J., Masmoudi N.: Well-posedness of compressible Euler equations in a physical vacuum. Comm. Pure Appl. Math. 68(1), 61–111 (2015). doi:10.1002/cpa.21517 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lax P.D.: Development of singularities of solutions of nonlinear hyperbolic partial differential equations. J. Math. Phys. 5, 611–613 (1964)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lindblad H.: Well-posedness for the linearized motion of a compressible liquid with free surface boundary. Comm. Math. Phys. 236(2), 281–310 (2003). doi:10.1007/s00220-003-0812-x ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lindblad H.: Well-posedness for the linearized motion of an incompressible liquid with free surface boundary. Comm. Pure Appl. Math. 56(2), 153–197 (2003). doi:10.1002/cpa.10055 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lindblad H.: Well posedness for the motion of a compressible liquid with free surface boundary. Comm. Math. Phys. 260(2), 319–392 (2005). doi:10.1007/s00220-005-1406-6 ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lindblad H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. 162(1), 109–194 (2005). doi:10.4007/annals.2005.162.109 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu T.P.: Compressible flow with damping and vacuum. Jpn. J. Ind. Appl. Math. 13(1), 25–32 (1996). doi:10.1007/BF03167296 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Majda, A.: Vorticity and the mathematical theory of incompressible fluid flow. Comm. Pure Appl. Math. 39(S, suppl.), S187–S220 (1986). doi:10.1002/cpa.3160390711. Frontiers of the Mathematical Sciences (New York, 1985)
  20. 20.
    Makino T., Ukai S., Kawashima S.: Sur la solution à support compact de l’équations d’Euler compressible. Jpn. J. Appl. Math. 3(2), 249–257 (1986). doi:10.1007/BF03167100 CrossRefMATHGoogle Scholar
  21. 21.
    Makino, T., Ukai, S., Kawashima, S.: On compactly supported solutions of the compressible Euler equation. In: Recent Topics in Nonlinear PDE, III (Tokyo, 1986), North-Holland Mathematics Studies, vol. 148, pp. 173–183. North-Holland, Amsterdam (1987). doi:10.1016/S0304-0208(08)72332-6
  22. 22.
    Rouchon P.: The Jacobi equation, Riemannian curvature and the motion of a perfect incompressible fluid. Eur. J. Mech. B Fluids 11(3), 317–336 (1992)MathSciNetMATHGoogle Scholar
  23. 23.
    Serre, D.: Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. Ann. Inst. Fourier 47(1), 139–153 (1997).
  24. 24.
    Serre D.: Expansion of a compressible gas in vacuum. Bull. Inst. Math. Acad. Sin. (N.S.) 10(4), 695–716 (2015)MathSciNetMATHGoogle Scholar
  25. 25.
    Sideris, T.C.: Formation of singularities in three-dimensional compressible fluids. Comm. Math. Phys. 101(4), 475–485 (1985).
  26. 26.
    Sideris T.C.: Spreading of the free boundary of an ideal fluid in a vacuum. J. Differ. Equ. 257(1), 1–14 (2014). doi:10.1016/j.jde.2014.03.006 ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130(1), 39–72 (1997). doi:10.1007/s002220050177 ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wu S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12(2), 445–495 (1999). doi:10.1090/S0894-0347-99-00290-8 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Santa BarbaraSanta BarbaraUSA

Personalised recommendations