Rotating Fluids with Small Viscosity

Global existence for the primitive equations with small anisotropic viscosity

“…We remark that, if ν ∼ ν ∼ ε α , then unlike the case of rotating fluids ( [35]) where the limit system is zero, the limit system of the primitive equations, when ε goes to zero, is the inviscid quasi-geostrophic system, which has no regularizing effect. So we have to estimate the H σ -norm, with σ > 5 2 , of the solution of the quasi-geostrophic system.…”

“…The idea is to carefully study the dependence with respect to the cut-off radii r and R of the constant C r,R arising in the Strichartz estimates in order to get an estimate which is able to absorb the blowing-up term 1 ε α . The goal of this paper is to extend the result obtained in [35] to the primitive equations (AP E ε ) and to show global existence of strong solutions of (AP E ε ) for large data when ε is small enough, in the case where ν = ε α , ν = ρε α for α ≤ α 0 and without any particular assumption on ρ. We will adapt the computations of eigenvalues and eigenvectors of the linearized primitive equations developed by F.Charve in [8], [7], [9], and [10] in the isotropic case to the anisotropic case.…”

“…Indeed, by using the method of [15], in the energy estimates, we have to deal with the coefficient 1 ν = 1 ε α , which goes to infinity when ε goes to 0. Nevertheless, in [35], the second author of this paper proved the global existence of a unique strong solution of (NSC) with ν = ε α , for large data, when ε is small and α ≤ α 0 . The idea is to carefully study the dependence with respect to the cut-off radii r and R of the constant C r,R arising in the Strichartz estimates in order to get an estimate which is able to absorb the blowing-up term 1 ε α .…”

“…• The quasi-geostrophic part of the initial data is zero: the limit system is then zero, so we can use the idea of [35] with the adapted computations from [7] and [9] to prove a Strichartz-type estimate which will help to absorb the blow-up term and then prove the global existence of strong solutions of (AP E ε ) for large data when ε is small. This is the case considered in this paper.…”

“…We will only give a sketch of the proof, since it is very close to the ones of [15] and [35] for example. The difference is that, like in [8] and [9], we have to be careful with the asymptotic expansions in order not to get a negative power of ε.…”