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The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory. (English) Zbl 0995.35051

Regularity globally in time of the three-dimensional viscous Camassa-Holmes (Navier-Stokes-\(\alpha\)) equations is proven. Estimates of the Hausdorff and fractal dimension of their global attractor are derived. It is shown that these equations can be regarded as a closure model for the Reynolds averaged Navier-Stokes equations.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76F02 Fundamentals of turbulence
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