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

By Ciprian Foias, Darryl D. Holm, Edriss S. Titi. 2001

452 citations recorded by the source corpus.

We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-α) 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, ℓ∈, 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/ℓ∈)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-α 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-α model to the NSE by proving a convergence theorem, that as the length scale α1 tends to zero a subsequence of solutions of the NS-α equations converges to a weak solution of the three dimensional NSE.

Subfields: Applied Mathematics, Computational Mechanics, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/6fc8e78472491b5c1537483f176334b02f5172ed

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Citations in the corpus

On questions of decay and existence for the viscous Camassa–Holm equations (2006, unknown): “Existence of Solutions for the VCHE Existence and uniqueness of solutions for the VCHE on periodic domains in three dimensions was proved first in [10] using the Galerkin method.”
Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains (2001, methodology): “Foias et al. (2001) have shown that as ¬ !”
On a well-posed turbulence model (2005, unknown): “The model (6) and Camassa-Holm model ([5]) have a similar energy balance and both satisfy a limit-consistency result.”
Variational principles for stochastic fluid dynamics (2014, methodology): “To guarantee that the resulting stochastic partial differential equations (SPDEs) are well-posed, one may need to mollify the Lagrangian to ensure analytical control of these gradients along the lines followed for the NavierStokes-alpha turbulence models [11, 12].”
A modified-Leray-α subgrid scale model of turbulence (2006, methodology): “This can be done along the lines of VCHE or NS-α in [15], or as has been observed for the Leray-α model in [9].”
On the existence and uniqueness of solutions to stochastic three-dimensional Lagrangian averaged Navier–Stokes equations (2006, methodology): “To be more precise, we study existence and uniqueness of solution for the 3D-Lagrangian Averaged Navier-Stokes (LANS-α) equations, also called viscous Camassa-Holm equations by some authors (see Foias et al. (2002) and the references therein), with homogeneous Dirichlet boundary condition in a bound”