Global Well-Posedness and Asymptotics for a Geophysical Fluid System

Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

“…In recent years, there were some mathematicians who considered the existence of strong solutions for the three-dimensional viscous primitive equations of large-scale atmosphere and ocean (see, e.g., [4,[9][10][11]18,20,33,34] and references therein). In [18], Guillén-González et al obtained the global existence of strong solutions to the primitive equations of large-scale ocean by assuming that the initial data are small enough, and also proved the local existence of strong solutions to the equation for all initial data.…”

Global existence for the primitive equations with small anisotropic viscosity

“…• 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.…”

“…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.…”

“…In [8], [9] and [10], in the case of constant isotropic viscosity and thermal diffusivity, the first author of this paper proved, for various initial data, that the limit system of the primitive equations is the quasi-geostrophic system (QG) when ε goes to zero. In [7], he considered strong solutions and showed that the oscillating part goes to zero, whereas the quasi-geostrophic part goes to the unique global solution of the 3D quasi-geostrophic system whose initial data is the quasi-geostrophic part of the initial data. As in [21], [24] and [14], the methods used in [7] are given by the study of spectral properties of the matrix −L + 1 ε PA, where P denotes the Leray projection of L 2 onto the subspace of divergence free vector fields.…”

“…In [7], he considered strong solutions and showed that the oscillating part goes to zero, whereas the quasi-geostrophic part goes to the unique global solution of the 3D quasi-geostrophic system whose initial data is the quasi-geostrophic part of the initial data. As in [21], [24] and [14], the methods used in [7] are given by the study of spectral properties of the matrix −L + 1 ε PA, where P denotes the Leray projection of L 2 onto the subspace of divergence free vector fields.…”