Stabilization and limiting amplitude of the solution of the Cauchy problem for nonhomogeneous Sobolev systems (Q1819281)

The authors consider systems \[ (*)\quad \frac{\partial \vec v}{\partial t}-[\vec v,{\vec \omega}]+\nabla p=F(x,t),\quad div \vec v=0\quad in\quad D,\quad \vec v(x,0)=\vec v_ 0(x),\quad div \vec v_ 0=0\quad on\quad R^ 3, \] where \(D=\{x,t:\) \(x\in R^ 3\), \(t>0\}\), \([v_ 1,v_ 2,v_ 3]=\vec v=\vec v(x,t)\) represents the velocity, \(p=p(x,t)\) represents the pressure, \({\vec \omega}=[0,0,\omega]\) is the Coriolis parameter, [.,.] denotes the vector product. First, the case \(F(x,t)=f(x)\) is considered. The stabilization of the solution of (*) to the solution of the corresponding steady-state system is proved. Moreover, under certain assumptions, the stabilization to the unique solution of the steady state problem is shown and its velocity is given. Further, the case \(F(x,t)=f(x)e^{i\lambda t}\) is studied, where \(\lambda\) is a real parameter. It is proved that the solution is stabilized to the periodic one. The velocity of this stabilization and the limit amplitude are given under the assumption \(\omega^ 2\neq \lambda^ 2\). If \(\omega^ 2=\lambda^ 2\) then the solution tends to infinity (the resonance case).