Differential properties of the solution to the first mixed boundary value problem for the Sobolev system (Q2719296)

The authors study the asymptotic properties of solutions \(({\mathbf v},P)\) to the Sobolev system NEWLINE\[NEWLINE{\mathbf v}_t-[{\mathbf v},{\mathbf w}]+ \nabla P= 0,\text{ div }{\mathbf v}= 0, t>0, x\in g\subseteq E_3, {\mathbf w}= (0,0,1), {\mathbf v}|_{t=0}={\mathbf v}^0, P|_{\partial g}= 0,NEWLINE\]NEWLINE for \(t\to \infty\), where \(g\) is a bounded region with a smooth boundary \(\partial g\). They found that for sufficiently smooth initial conditions the following two cases are possible:NEWLINENEWLINENEWLINE(1) For an arbitrary interior point \(x_0\in g\) the functions \(D^2_t P(x_0, t)\), \(D^3_t v_i(x_0, t)\), \(i= 1,2,3\), considered as functions of \(t\) in \((0,\infty)\), are integrable on \((0,\infty)\) and monotonously tending to \(0\) for \(t\to\infty\).NEWLINENEWLINENEWLINE(2) The derivative of these functions with respect to \(t\) is oscillatory in every neighborhood of \(t=\infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].