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

V. A. Solonnikov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 14, Tome 115 (1982), pp. 251-263

Résumé

In unbounded domains $\Omega$ of the three-dimensional Euclidean space, having several exits $\Omega_i$ at infinity of a sufficiently general form, one finds the solution $\vec v(x)$ of the stationary Navier–Stokes system, equal to zero on the boundary of the domain $\Omega,$ having arbitrary flow rates $\alpha_i$ through each exit $\Omega_i$, $i=1,\dots,m$ ($\sum_{i=1}^m\alpha_i=0$), and having an unbounded Dirichlet integral $\int_\Omega|\vec v_x|^2\,dx=+\infty$. One gives sufficient conditions for the existence of a solution.