## 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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