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Energy Conservation in Two-dimensional Incompressible Ideal Fluids

Cheskidov, A. ; Filho, M. C. Lopes ; Lopes, H. J. Nussenzveig ; Shvydkoy, R.

Communications in mathematical physics, 2016-11, Vol.348 (1), p.129-143 [Periódico revisado por pares]

Berlin/Heidelberg: Springer Berlin Heidelberg

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  • Título:
    Energy Conservation in Two-dimensional Incompressible Ideal Fluids
  • Autor: Cheskidov, A. ; Filho, M. C. Lopes ; Lopes, H. J. Nussenzveig ; Shvydkoy, R.
  • Assuntos: Classical and Quantum Gravitation ; Complex Systems ; Energy conservation ; Mathematical and Computational Physics ; Mathematical Physics ; Navier-Stokes equations ; Physics ; Physics and Astronomy ; Quantum Physics ; Relativity Theory ; Theoretical
  • É parte de: Communications in mathematical physics, 2016-11, Vol.348 (1), p.129-143
  • Descrição: This note addresses the issue of energy conservation for the 2D Euler system with an L p -control on vorticity. We provide a direct argument, based on a mollification in physical space, to show that the energy of a weak solution is conserved if ω = ∇ × u ∈ L 3 2 . An example of a 2D field in the class ω ∈ L 3 2 - ϵ for any ε > 0 , and u ∈ B 3 , ∞ 1 / 3 (Onsager critical space, see Shvydkoy in Discr Contin Dyn Syst Ser S 3(3):473–496, 2010 ) is constructed with non-vanishing energy flux. This demonstrates sharpness of the kinematic argument, which does not differentiate between 2D and 3D, and requires Onsager’s regularity control on the solution. Next, we show that for physically realizable solutions there is a mechanism preventing the anomalous dissipation in 2D that does not require such a control. Namely, we prove that any solution to the Euler equations produced via a vanishing viscosity limit from the Navier–Stokes equations, with ω ∈ L p , for p >  1, conserves energy.
  • Editor: Berlin/Heidelberg: Springer Berlin Heidelberg
  • Idioma: Inglês
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  • Realização: ABCD-USP USP   Logos de Redes Sociais: Freepik

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