Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier–Stokes equations

We consider the problem of finding the restrictions on the domain $\Omega\subset R^n$, $n=2,3$, under which the space $\overset{\hat\circ}J{}^1_2(\Omega)$ of the solenoidal vector fields from $\overset{\circ}W{}^1_2(\Omega)$ coincides with the space $\overset{\circ}J{}^1_2(\Omega)$, the closure in $W_2^1(\Omega)$ of the set of all solenoidal vectors from $\dot C^\infty(\Omega)$. We give domains $\Omega\subset R^n$, for which the factor space $\overset{\hat\circ}J{}^1_2(\Omega)/\overset{\circ}J{}^1_2(\Omega)$ has a finite nonzero dimension. A similar problem is considered for the spaces of solenoidal vectors with a finite Dirichlet integral. Based on this, one compares two generalized formulations of boundary-value problems for the Stokes and Navier–Stokes systems. The following auxiliary problems are studied: 1) $\operatorname{div}\vec{u}=\varphi$, $\vec{u}|_{\partial\Omega}=0$; 2) $\operatorname{div}\vec{u}=0$, $\vec{u}|_{\partial\Omega}=\vec{\alpha}$; 3) $\operatorname{grad}p=\sum\limits^n_{k=1}\dfrac{\partial\vec{R}_k}{\partial x_k}+\vec{f}$.