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Modified dissipativity for a non linear evolution equation arising in turbulence

Part of the Lecture Notes in Mathematics book series (LNM,volume 565)

Abstract

We are concerned with the global (in time) regularity properties of the Burgers MRCM equation, which arises in the theory of turbulence (with α = 1)

where U(t,·) is of positive type and where the dissipativity α is a nonnegative real number. It is shown that for arbitrary ν > 0 and ɛ > 0, there exists a global solution in L[0,∞ (ℝ)]. If ν > 0 and α > αcr = 1/2, smoothness of initial data persists indefinitely. If 0 < α < αcr, there exist positive data-dependent constants ν1(α) et ν2(α) such that indefinite persistence of regularity holds for ν > ν1(α), whereas for 0 < ν < ν2 (α) the second spatial derivative at the origin blows up after a finite time. It is conjectured that with a suitable choice of αcr, similar results hold for the Navier-Stokes equation.

Keywords

  • Initial Data
  • Dissipative Term
  • Nonnegative Real Number
  • Sobolev Norm
  • Global Regularity

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  • BRAUNER, C.M., PENEL, P. and TEMAM, R. (1974) C.R. Acad. Sc. Paris, A.279, 65 and 115.

    MathSciNet  MATH  Google Scholar 

  • BRISSAUD, A., FRISCH, U., LEORAT, J., LESIEUR, M., MAZURE, A., POUQUET, A., SADOURNY, R., and SULEM, P.L., (1973), Ann. Geophys., 29, 539.

    Google Scholar 

  • BURGERS, J.M. (1940), Proc. Roy. Netherl. Acad., 43, 2.

    MathSciNet  Google Scholar 

  • FOIAS, C. and PENEL, P. (1975), C.R. Acad. Sc. Paris, A.280, 629.

    MathSciNet  MATH  Google Scholar 

  • FRISCH, U. (1974), Proceedings of the Conference on Prospect for Theoretical Turbulence Research NCAR, Boulder, Colorado.

    Google Scholar 

  • FRISCH, U., LESIEUR, M., and BRISSAUD, A. (1974), J. Fluid Mech., 65, 145.

    CrossRef  MATH  Google Scholar 

  • GOULAOUIC, C. and BAOUENDI, S. (1975), Private Communication.

    Google Scholar 

  • HERRING, J.R. and KRAICHNAN, R.H. (1972) in Statistical Models and Turbulence, p. 148, Springer.

    Google Scholar 

  • KATO, T. (1972), J. Funct. Anal., 9, 296.

    CrossRef  MATH  Google Scholar 

  • KOLMOGOROV, N.A. (1941), C.R. Acad. Sc. URSS, 30, 301.

    Google Scholar 

  • KOLMOGOROV, N.A. (1962), J. Fluid Mech., 12, 82.

    CrossRef  MathSciNet  Google Scholar 

  • KRAICHNAN, R.H. (1961), J. Math. Phys., 2, 124; also, erratum 3, 205 (1962).

    CrossRef  MathSciNet  MATH  Google Scholar 

  • KRAICHNAN, R.H. (1974), J. Fluid Mech., 62, 305.

    CrossRef  MathSciNet  MATH  Google Scholar 

  • KRUZKOV, S.N. (1970), First Order Quasilinear Equations in Several Independent Variables. Math. USSR Sbornik, vol. 10, 217.

    CrossRef  Google Scholar 

  • LADYZENSKAYA, O.A. (1963), A Mathematical Theory of Viscous Incompressible Flow. (First edition, Gordon and Breach, New-York).

    Google Scholar 

  • LESIEUR, M. (1973), Thesis, University of Nice.

    Google Scholar 

  • LESIEUR, M. and SULEM, P.L. (1975), Les Equations Spectrales en Turbulence Homogène et Isotrope. Quelques Résultats Théoriques et Numériques. Proc. of this Conference.

    Google Scholar 

  • LIONS, J.L. (1969), Quelques Méthodes de Résolution des Problèmes aux Limites non Lineaires. Dunod-Gauthier-Villars.

    Google Scholar 

  • ORSZAG, S.A. (1975), Lectures on the Statistical Theory of Turbulence. Proceedings of the 1973 Les Houches Summer School of Theoretical Physics.

    Google Scholar 

  • PENEL, P. (1975), Thesis, University of Paris-Sud, Orsay.

    Google Scholar 

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Bardos, C., Penel, P., Frisch, U., Sulem, P.L. (1976). Modified dissipativity for a non linear evolution equation arising in turbulence. In: Temam, R. (eds) Turbulence and Navier Stokes Equations. Lecture Notes in Mathematics, vol 565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091444

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  • DOI: https://doi.org/10.1007/BFb0091444

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08060-2

  • Online ISBN: 978-3-540-37516-6

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