Integrability and regularity of 3D Euler and equations for uniformly rotating fluids

Global existence for the primitive equations with small anisotropic viscosity

“…In order to conclude the classical boostrap method, we required that the parameters satisfy The greatest possible value for the parameter α is 1 11 and the greatest we want to choose α (nearest α < 1 11 ), the greatest has to be η which means we must require more regularity on the initial data. Typically, choose α < 1 11 , then choose η > 0 such that α < 1 11+ 44 η , which implies that α η < 1−11α 44 . Finally, use (5.23) and choose β between those two values (γ = 1).…”

“…Technical problems forced α < 1 11 but the question of the optimal value of α is open (see the next paragraph).…”

“…For any real number s > 1 2 , there exists a constant C such that for any vector fields u and v, where u is divergence free,…”

The Kolmogorov–Obukhov Statistical Theory of Turbulence

“…If the initial data had the symmetry of the swirl then the deterministic problem would be two-dimensional and the global existence of the two-dimensional Navier-Stokes equation is well known. It is also well-known that if the initial data is close to such a two-dimensional flow then global existence can be extened to this case also, see [2,3], for another such example.…”

“…We know that the above invariant measure is unique for the strong swirl [12] and strong rotation [2,3] but it depends on u, and its uniqueness for general turbulent flows depends on the uniqueness of u.…”