Mathematical Analysis of the Primitive Equations with Rotation
Abstract
Large planetary scale dynamics of the oceans and the atmosphere is governed by the primitive equations (PEs). It is well-known that the $3D$ viscous PEs is globally (in time) well-posed in Sobolev spaces. On the other hand, the inviscid primitive equations (IPEs) without rotation is known to be ill-posed in all Sobolev spaces, and some of its smooth solutions can form singularities in finite time. In this thesis, the above results are extended in the presence of rotation (Coriolis force). More specifically, certain finite-time blowup solutions to the IPEs with rotation are constructed, and it is established that the IPEs with rotation is ill-posed in the sense that the perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in all Sobolev spaces, and is linearly ill-posed in Gevrey class of order $s>1$.
Although the IPEs is ill-posed in Sobolev spaces and Gevrey class of order $s>1$, it is shown in this thesis that the $3D$ IPEs is locally (in time) well-posed in the space of analytic functions, i.e., the Gevrey class of order $s=1$, for a short interval of time that is independent of the rotation rate. By the comparison between the $3D$ IPEs and the $2D$ Euler equations, one can establish the long-time existene of solutions to the $3D$ IPEs provided the analytic norm of the initial baroclinic mode is small enough, while the initial barotropic mode can be large. Moreover, one can show that, in the case of ``well-prepared'' analytic initial data (only the Sobolev norm of the baroclinic mode is small depending on the rotation rate, while the analytic norm can be large), the regularizing effect of the Coriolis force by providing a lower bound for the life-span of the solutions which grows toward infinity with the rotation rate. The latter is achieved by a delicate analysis of a simple limit resonant system whose solution approximates the corresponding solution of the $3D$ IPEs with the same initial data.
The PEs with only vertical viscosity (also called the hydrostatic Navier-Stokes equations) is believed to be ill-posed in Sobolev spaces. To overcome the potential ill-posedness, some weak dissipations are introduced in the horizontal directions, which are the linear (Rayleigh-like friction) damping terms. With these damping terms, it is established that this system is locally well-posed with general Sobolev initial data and globally well-posed with small Sobolev initial data. In order to study the possible finite-time blow-up and to give a reliable numerical regularization, it is proposed to study the Voigt $\alpha$-regularization of this model, which is an inviscid regularization. One is able to establish the global well-posedness of the regularized model for arbitrary Sobolev initial data. In addition, it is shown that the solutions of the regularized model converge to those of the original model on the interval of the existence of the latter, as $\alpha \rightarrow 0$. Based on this convergence result, a blowup criterion of the original model is established.
Citation
Lin, Quyuan (2021). Mathematical Analysis of the Primitive Equations with Rotation. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195650.