Three dimensional flux problem for the Navier - Stokes equations

Solvability of the flux problem for the Navier - Stokes equations has been proven by J. Leray (1933) under an additional condition of zero flux through each connected component of the flow domain boundary. He used arguments from contradiction was by contradiction and did not give a priory estimate of solution. This estimate was obtained by E. Hopf (1941) under the same condition concerning fluxes. The following problem is open up to now: if exists a solution of the flux problem, when only the necessary condition of total zero flux is satisfied? For small fluxes values, solvability of three dimensional problem was established independently by H. Fujita and R. Finn (1961). H. Fujita and H. Morimoto (1995) proved existence theorem for flows, which are close to potential ones). M.V. Korobkov, K. Pileckas and R. Russo (2011 - 2015) gave the positive solution of the flux problem for planar and axially symmetric flows without restrictions on the flux values. The present paper is devoted to consideration of flux problem in the domain of a spherical layer type. We obtained an a priori estimate of solution under following additional conditions: the flow has a symmetry plane; the flux through the inner domain boundary is positive. This estimate implies solvability of the problem.