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Regularity of Euler Equations for a Class of Three-Dimensional Initial Data

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Trends in Partial Differential Equations of Mathematical Physics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 61))

Abstract

The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at t = 0.

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References

  1. V.I. Arnold (1965), Small denominators. I. Mappings of the circumference onto itself, Amer. Math. Soc. Transl. Ser. 2, 46, p. 213–284.

    Google Scholar 

  2. V.I. Arnold and B.A. Khesin (1997), Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125, Springer.

    Google Scholar 

  3. A. Babin, A. Mahalov, and B. Nicolaenko (1997), Global regularity and integrability of 3D Euler and Navier-Stokes equations for uniformly rotating fluids, Asymptotic Analysis, 15, p. 103–150.

    MATH  MathSciNet  Google Scholar 

  4. A. Babin, A. Mahalov and B. Nicolaenko (1999), Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48, No. 3, p. 1133–1176.

    MATH  MathSciNet  Google Scholar 

  5. A. Babin, A. Mahalov and B. Nicolaenko (2001), 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50, p. 1–35.

    MATH  MathSciNet  Google Scholar 

  6. C. Bardos, F. Golse, A. Mahalov and B. Nicolaenko (2004), Long-time regularity of 3D Euler equations with initial data characterized by uniformly large vorticity in cylindrical domains, to appear.

    Google Scholar 

  7. J.T. Beale, T. Kato and A. Majda (1984), Remarks on the breakdown of smooth solutions for the 3D Euler equations, Commun. Math. Phys., 94, p. 61–66.

    Article  MATH  MathSciNet  Google Scholar 

  8. A.S. Besicovitch (1954), Almost Periodic Functions, Dover, New York.

    Google Scholar 

  9. J.P. Bourguignon and H. Brezis (1974), Remark on the Euler equations, J. Func. Anal., 15, p. 341–363.

    Article  MATH  MathSciNet  Google Scholar 

  10. E.B. Bykhovskii (1957), Solutions of problems of mixed type of the Maxwell’s equations for ideally conducting boundaries, Vestnik Leningrad Univ., 13, p. 50–66.

    Google Scholar 

  11. E.B. Bykhovskii and N.V. Smirnov (1960), On orthogonal expansions in spaces of square integrable vector functions and operators of vector analysis, Trudy Math. Inst. V.A. Steklov, Special Volume on Mathematical Problems in Hydrodynamics and Magnetohydrodynamics, 59, p. 5–36.

    Google Scholar 

  12. E. Cartan (1922), Sur les petites oscillations d’une masse fluide, Bull. Sci. Math., 46, p. 317–352 and p. 356–369.

    Google Scholar 

  13. C. Corduneanu (1968), Almost periodic Functions, Wiley-Interscience, New York.

    MATH  Google Scholar 

  14. G. Duvaut and J.L. Lions (1976), The Inequalities in Mechanics and Physics, Springer-Verlag.

    Google Scholar 

  15. K. Friedrichs (1955), Differential forms on Riemannian manifolds, Comm. Pure Appl. Math, 8, No. 2.

    Google Scholar 

  16. T. Kato (1972), Nonstationary flows of viscous and ideal fluids in R 3, J. Func. Anal., 9, p. 296–305.

    Article  MATH  Google Scholar 

  17. N.D. Kopachevsky and S.G. Krein (2001),Operator Approach to Linear Problems of Hydrodynamics, Operator Theory: Advances and Applications, 128, Birkhäuser Verlag.

    Google Scholar 

  18. O.A. Ladyzhenskaya ed. (1960), Trudy Math. Inst. V.A. Steklov, Special Volume on Mathematical Problems in Hydrodynamics and Magnetohydrodynamics, 59.

    Google Scholar 

  19. O.A. Ladyzhenskaya (1968), On the unique global solvability of the 3D Cauchy problem for the Navier-Stokes equations under conditions of axisymmetry, Boundary Value problems of Math. Phys. and Related Problems No. 2, Zapiski Seminarov LOMI, 7, p. 155–177.

    MATH  Google Scholar 

  20. O.A. Ladyzhenskaya (1969),The Mathematical Theory of Viscous Incompressible Flow, 2nd ed., Gordon and Breach, New York.

    MATH  Google Scholar 

  21. O.A. Ladyzhenskaya and V.A. Solonnikov (1960), Solution of some nonstationary problems of magnetohydrodynamics for viscous incompressible fluids, Trudy Math. Inst. V.A. Steklov, Special Volume on Mathematical Problems in Hydrodynamics and Magnetohydrodynamics, 59, p. 115–173.

    Google Scholar 

  22. J. Leray (1934), Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, p. 193–248.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Mahalov, S. Leibovich and E.S. Titi (1990), Invariant helical subspaces for the Navier-Stokes Equations, Arch. for Rational Mech. and Anal., 112, No. 3, p. 193–222.

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Mahalov and B. Nicolaenko (2003), Global solvability of the three-dimensional Navier-Stokes equations with uniformly large initial vorticity, Russian Math. Surveys, 58:2, p. 287–318.

    Article  MathSciNet  Google Scholar 

  25. H. Poincaré (1910), Sur la précession des corps déformables, Bull. Astronomique, 27, p. 321–356.

    Google Scholar 

  26. J.V. Ralston (1973), On stationary modes in inviscid rotating fluids, J. Math. Anal. and Appl., 44, p. 366–383.

    Article  MATH  MathSciNet  Google Scholar 

  27. S. Schochet (1994), Resonant nonlinear geometric optics for weak solutions of conservation laws, J. Diff. Eq., 113, p. 473–503.

    Article  MATH  MathSciNet  Google Scholar 

  28. S. Schochet (1994), Fast singular limits of hyperbolic PDE’s, J. Diff. Eq., 114, p. 476–512.

    Article  MATH  MathSciNet  Google Scholar 

  29. S.L. Sobolev (1954), Ob odnoi novoi zadache matematicheskoi fiziki, Izvestiia Akademii Nauk SSSR, Ser. Matematicheskaia, 18, No. 1, p. 3–50.

    MathSciNet  Google Scholar 

  30. V.A. Solonnikov (1972), Overdetermined elliptic boundary value problems, Boundary Value problems of Math. Phys. and Related Problems No. 5, Zapiski Seminarov LOMI, 21, p. 112–158.

    MathSciNet  Google Scholar 

  31. E.M. Stein (1970), Singular integrals and differentiability properties of functions, Princeton University Press.

    Google Scholar 

  32. W. Wolibner (1933), Un théorème sur l’existence du mouvement plan d’un fluide parfait, homogène et incompressible, pendant un temps infiniment long, Mat. Z., 37, p. 698–726.

    Article  MathSciNet  Google Scholar 

  33. V.I. Yudovich (1963), Non stationary flow of an ideal incompressible liquid, Zb. Vych. Mat., 3, p. 1032–1066.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287-1804, USA

    A. Mahalov & B. Nicolaenko

  2. U.M.R. of Mathematics, University of Paris-7, F-75251, Paris Cedex-05

    C. Bardos & F. Golse

  3. Laboratoire J.L. Lions, University of Paris-6, France

    C. Bardos & F. Golse

Authors
  1. A. Mahalov
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  2. B. Nicolaenko
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  3. C. Bardos
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  4. F. Golse
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Editor information

Editors and Affiliations

  1. Universidade de Lisboa/CMAF, Av. Prof. Gama Pinto, 2, 1649-003, Lisboa, Portugal

    José Francisco Rodrigues

  2. Steklov Institute of Mathematics, Russian Academy of Science, 27, Fontanka, St. Petersburg, 191023, Russia

    Gregory Seregin

  3. Departamento de Matemática, Universidade de Coimbra/FCT, Apartado 3008, 3001-454, Coimbra, Portugal

    José Miguel Urbano

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© 2005 Birkhäuser Verlag Basel/Switzerland

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Mahalov, A., Nicolaenko, B., Bardos, C., Golse, F. (2005). Regularity of Euler Equations for a Class of Three-Dimensional Initial Data. In: Rodrigues, J.F., Seregin, G., Urbano, J.M. (eds) Trends in Partial Differential Equations of Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 61. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7317-2_13

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