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Modified dissipativity for a non linear evolution equation arising in turbulence

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Turbulence and Navier Stokes Equations

Part of the book series: Lecture Notes in Mathematics ((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.

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Author information

Authors and Affiliations

  1. Dept. de Mathématique IMSP, Université de Nice, France

    C. Bardos

  2. Dept. de Mathématique, Université de Paris-Sud, Orsay

    P. Penel

  3. Centre National de la Recherche Scientifique Observatoire de Nice, France

    U. Frisch & P. L. Sulem

Authors
  1. C. Bardos
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  2. P. Penel
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  3. U. Frisch
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  4. P. L. Sulem
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Roger Temam

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© 1976 Springer-Verlag Berlin · Heidelberg

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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

  • eBook Packages: Springer Book Archive

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