Global Well--posedness of the Three-dimensional Viscous Primitive Equations of Large Scale Ocean and Atmosphere Dynamics
Chongsheng Cao, Edriss S. Titi
TL;DR
The paper addresses the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large-scale ocean and atmosphere dynamics, in a cylindrical domain. The approach splits the velocity into a horizontal mean $\overline{v}$ and a fluctuation $\widetilde{v}$, derives coupled mean/fluctuation equations with homogeneous boundary conditions after integrating the vertical momentum to express $w$, and (under simplifying assumptions) takes $\tau=0$ and $T^*=0$ to focus on core estimates. The paper establishes a priori bounds on $L^2$, $L^6$, and $H^1$-type quantities for $(v,T)$ and their derivatives, including $L^2$ energy estimates for $(T,v)$, a Gronwall-type bound for $\| ilde{v}\|_{L^6}$ via testing with $|\tilde{v}|^4 \tilde{v}$, and bounds for $\nabla \overline{v}$, $v_z$, $\nabla v$, and $\nabla T$. Together, these a priori estimates yield global existence, uniqueness, and continuous dependence on initial data for strong solutions, demonstrating global regularity of the primitive equations under the considered setting and indicating robustness with respect to boundary conditions and potential salinity effects.
Abstract
In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics.