Solutions of the stationary Navier-Stokes system of equations with an infinite Dirichlet integral

Abstract

In unbounded domains Ω of the three-dimensional Euclidean space, having several exits Ωi at infinity of a sufficiently general form, one finds the solutions\(\vec \upsilon (x)\) of the stationary Navier-Stokes system, equal to zero on the boundary of the domain Ω, having arbitrary flow rates di through each exit Ωi, i=1,...,\(m\left( {\sum\limits_{i = 1}^m {d_i = 0} } \right)\), and having an unbounded Dirichlet integral\(\smallint _\Omega \left| {\vec \upsilon _x } \right|^2 dx = + \infty\). One gives sufficient conditions for the existence of a solution.