The Three Dimensional Viscous Camassa-Holm Equations, and Their Relation to the Navier-Stokes Equations and Turbulence Theory

Abstract

We show here the global, in time, regularity of the three dimensional viscous Camassa-Holm (Lagrangian Averaged Navier-Stokes-alpha) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, \ell_ε, as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the long-time behavior of the solutions to these equations is bounded from above by (L/\ell_{epsilon})^3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau-Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NS-alpha equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier-Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NS-alpha model to the NSE by proving a convergence theorem, that as the length scale alpha tends to zero a subsequence of solutions of the NS-alpha equations converges to a weak solution of the three dimensional NSE.

arXiv comment26 pages, no figures. Dedicated to Roger Temam on the Occasion of his 60-th birthday. To appear in Journal of Dynamics and Differential Equations

References in corpus (4)

• An integrable shallow water equation with peaked solitons
• The Camassa-Holm equations as a closure model for turbulent channel flow
• Direct Numerical Simulations of the Navier-Stokes Alpha Model
• The anisotropic averaged Euler equations

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• On 3D Lagrangian Navier-Stokes $\alpha$ model with a Class of Vorticity-Slip Boundary conditions
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