Regularity results for a Navier-Stokes type problem related to oceanography (Q1370796)

The paper investigates a model of an ocean flow in tropical zones. The authors study the perturbation \((u,p)\) of a given mean flow \((u_0,p_0)\). This perturbation satisfies the system of equations \(\frac{\partial u}{\partial t}+(u\cdot\nabla)u +(u_0\cdot\nabla)u+ (u\cdot\nabla)u_0+F\times u\), \(-\nu_h\Delta' u-\frac\partial{\partial z} (\nu_v \frac{\partial u}{\partial z}) +\nabla p=0\), \(\text{div}u=0\) in \(\Omega\times(0,T),\quad \Omega=\{(x,y,z):\;0<x<\ell_1,\) \(-\ell_2<y<\ell_2,\;-H<z<0\}\). Here \(F\) is the Coriolis force, \(\nu_h=const\) is the horizontal eddy viscosity, the function \(\nu_v\) is the vertical eddy viscosity, \(\Delta'=\frac{\partial^2}{\partial x^2}+\frac{\partial^2} {\partial y^2}\). The system is supplemented by the initial condition \(u|_{t=0}\) and by the boundary conditions \(u|_{x=0}= u|_{x=\ell_1}\) (periodicity in \(x\)-direction), \(u_2=0\), \(\frac{\partial u_1}{\partial y}=0\), \(\frac{\partial u_3}{\partial y}=0\) on planes \(y=-\ell_2\), \(y=\ell_2\), \(u|_{z=-H}=0\) (perturbation vanishes at the bottom), \(u_3=0\), \(\frac{\partial u_1}{\partial z}=-\frac{f_1}{\nu_v}\), \(\frac{\partial u_2}{\partial z}= -\frac{f_2}{\nu_v}\), on the surface \(z=0\), where \(u=(u_1,u_2,u_3)\), and \(f=(f_1,f_2)\) is the perturbation of the wind stress. Using the method of linearization, the authors prove that \(u\in L^2((0,T), H^2(\Omega))\), \(p\in L^2((0,T), H^1(\Omega))\), provided \(f\) is sufficiently regular and satisfies compatibility relations.