On the Solvability of the Stokes and Navier-Stokes Problems in the Domains That Are Layer-Like at Infinity

Weak solutions to the Stokes and Navier-Stokes problems are proved to exist in domains which, outside a ball, coincide with the three-dimensional layer R-2 x (0, 1). Apart from solutions with the finite Dirichlet integral, solutions to the linear problem are constructed with a prescribed behavior at infinity such that the plane-parallel Poiseuille and Couette flows, the rotational flow. A solution to the nonlinear problem is found that drives a nonzero flux to infinity and becomes unique under the data smallness assumption. Estimates for weighted norms of the pressure are derived as well.