A variational principle for the nonstationary linear Navier-Stokes equations (Q1313497)

A boundary-value problem for the linear nonstationary Navier-Stokes system with time-averaged data is considered: \[ u_ t- \nu\Delta u+ \nabla p=F, \quad \text{div } u=0, \qquad x\in\Omega,\;t\in (0,T); \tag{1} \] \[ (2)\quad u=0,\;x\in \partial\Omega, \qquad (3)\quad \sum_{i=0}^ m \gamma_ i u(x,t_ i)= f(x),\;x\in\Omega. \] Here \(\Omega\subset \mathbb{R}^ n\), \(n\geq 2\) is a bounded domain with a smooth boundary. The functions \(F\) and \(f\) are given. The numbers \(\gamma_ i\), \(t_ i\), \(T\) are prescribed, \(0=t_ 0< t_ 1<\dots <t_ m= T\), \(\gamma_ 0=1\). The solvability and uniqueness to the problem (1), (2), (3) are established in a space \(H_{x,t}^{2,1}\) if the numbers \(\gamma_ i\) are sufficiently small. A similar result is obtained for a problem for which summation in (3) is exchanged by integration. The classical methods of functional analysis and Galerkin approximation are used.