Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids

Ekman layers of rotating fluids: The case of general initial data

“…We consider the following system of equations: (1.4) where Ω = T 2 ×]0, h[, T 2 is the periodic torus in x and y with periods of a 1 and a 2 , respectively, and ∆ x,y denotes the two-dimensional Laplace operator in the variables x and y. The vector e 3 is the unit vertical vector, F γ is an exterior force that belongs to L 1 (0, T ; L 2 (Ω) 3 ), and (1/ε)e 3 × u γ = (1/ε)(−u γ 2 , u γ 1 , 0) is the Coriolis force created by the rotation at the frequency ε −1 , which is assumed to be large. We also assume that u γ 0 converges strongly in L 2 (Ω) 3 to a function u 0 when (ε, ν) tends to (0, 0), η tends to 0 or is constant, and γ denotes (ε, ν, η).…”

“…The vector e 3 is the unit vertical vector, F γ is an exterior force that belongs to L 1 (0, T ; L 2 (Ω) 3 ), and (1/ε)e 3 × u γ = (1/ε)(−u γ 2 , u γ 1 , 0) is the Coriolis force created by the rotation at the frequency ε −1 , which is assumed to be large. We also assume that u γ 0 converges strongly in L 2 (Ω) 3 to a function u 0 when (ε, ν) tends to (0, 0), η tends to 0 or is constant, and γ denotes (ε, ν, η). In what follows, we take U = (u, p) = (u 1 , u 2 , u 3 , p), where we omit the γ if there is no ambiguity.…”

Long Time Stability for Solutions of a β‐Plane Equation

“…This rate of decay is not integrable, so obtaining good control of solutions to the nonlinear equation (1.4) for long periods of time is a challenge. However, it is the most decay one may hope for in a two-dimensional model, so despite being 2 More precisely, for data of size " smallness is guaranteed to hold up to a time of order jlog. "/j 1 " 2 .…”

“…Another important achievement in the study of both the rotating Euler equations and Navier-Stokes equations (in three dimensions) is the series of works of Babin-Mahalov-Nicolaenco [1][2][3][4], which focused on studying the case of fast rotation. They proved that as the speed of rotation increases (depending upon the initial data), the solution of the rotating Navier-Stokes equations becomes globally wellposed and the solution of the Euler equations exists for a longer and longer time.…”

“…Indeed, as we already observed in our previous work on the related SQG equation [10, remark 3.3], the smallness of solutions can be guaranteed for a longer period of time. 2…”