Existence and uniqueness of strong periodic solution of the evolution systems describing geophysical flow. I: In bounded domains (Q5943624)

This paper deals with the following evolution system describing geophysical flow in a bounded domain \(\mathbb{R}^N\) \((N=3,4)\), that is \[ \begin{cases} {\partial u\over \partial t}-\nu \Delta u+(u\cdot \nabla)u-{1\over\rho \mu}(B \cdot\nabla) B+{1\over 2\rho\mu} \nabla\bigl(|B|^2 \bigr)+ {1\over\rho} \nabla p=f(x,t),\\ {\partial B\over\partial t}-\lambda\Delta B+(u\cdot \nabla)B-(B\cdot\nabla) u+{1\over\mu} \nabla q=g(x,t),\\ \nabla\cdot u=0,\;\nabla\cdot B= 0, \end{cases} \tag{1} \] where \(u=u(x_1,\dots, x_N,t)=(u_1,\dots, u_N)\) and \(B= B(x_1, \dots,x_N,t)\) are the velocity vectors of Eulerian flow and magnetic fields respectively, \(p(x,t)\) and \(q(x,t)\) are pressures, \(f(x,t)\) and \(g(x,t)\) are volume forces, and \(\rho\) and \(\nu\) are the constants of density and viscosity of the flow, respectively, with \(\mu\) is a constant of magnetic permeability and \(\lambda={\eta\over\mu}\) with electrical resistivity. Under the assumption that the external forces \(f\) and \(g\) are sufficiently small and periodic, the authors prove both existence and uniqueness of a strong periodic solution of (1).