Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains

Abstract

In this paper, we study the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains. The LANS-α equations are able to accurately reproduce the large-scale motion (at scales larger than α > 0) of the Navier-Stokes equations while filtering or averaging over the motion of the fluid at scales smaller than α, an a priori fixed spatial scale.We prove the global well-posedness of weak H 1 solutions for the case of no-slip boundary conditions in three dimensions, generalizing the periodic-box results of [8]. We make use of the new formulation of the LANS-α equations on bounded domains given in [20] and [14], which reveals the additional boundary conditions necessary to obtain well-posedness. The uniform estimates yield global attractors; the bound for the dimension of the global attractor in 3D exactly follows the periodic box case of [8]. In 2D, our bound is α-independent and is similar to the bound for the global attractor for the 2D Navier-Stokes equations.1991 Mathematics Subject Classification. 35Q35. Key words and phrases. Lagrangian averaged Navier-Stokes, Turbulence, Model. 1 Some authors had previously referred to the LANS-α equations as the Viscous Camassa-Holm equations.

On THE STOCHASTIC 3d-Lagrangian AVERAGED NAVIER–STOKES Α-Model WITH FINITE DELAY

“…Finally, concerning the term Au − α∆(Au), we have that if u ∈ M 2 F t (0, T ; D(A)), then Au − α∆(Au) ∈ M 2 F t (0, T ; (H −2 (D)) 3 ). Reasoning again as in [8], and more precisely, by using Remark 4.3 from [8], it follows the uniqueness of p.…”

“…First, observe that I + αA is bijective from D(A) into H, and , and, as a consequence, ∂ t (u − α∆u) ∈ L 2 (Ω, F t , P ; H −1 (0, T ; (L 2 (D)) 3 ). Also, if u ∈ L 4 (Ω, F, P ; C([0, T ]; V )), and it is F t -progressively measurable, then F (t, u t ) ∈ M 4 F t (0, T ; (H −1 (D)) 3 ), and arguing as in [8], if follows G(t, u t ) ∈ L 4 (Ω, F t , P ; W −1,∞ (0, t; (L 2 (D) 3 ))), ∀ t ∈ [0, T ].…”

“…We will denote (•, •) and |•|, respectively, the scalar product and associated norm in (L 2 (D)) 3 , and by (∇u, ∇v) the scalar product in ((L 2 (D)) 3 ) 3 of the gradients of u and v. We consider the scalar product in (H 1 0 (D)) 3 defined by 10) and so D(A) is a Hilbert space with respect to the scalar product…”

“…Problem (1.1) in the deterministic case, i.e. when G = 0, and when F does not depend on u, has been studied in [3] (see also [10]). In the stochastic case but without delay, it has been analysed in [2].…”

On the viscous Camassa-Holm equations with fractional diffusion

“…For a more comprehensive history of the LANS-α equation, we refer the readers to [28] and the references therein. As for results in analysis, a handful of global existence or well-posedness results of the LANSα equation have been established in periodic boxes [17], in bounded domains and the whole space [15,2,1], and on Riemannian manifolds with boundaries [31]; decay of solutions in bounded domains and the whole spaces was also investigated in [2,1].…”