Three-dimensional flux problem for the Navier-Stokes equations.

*(Russian. English summary)*Zbl 1336.76012Summary: Solvabily of the flux problem for the Navier-Stokes equations has been proven by J. Leray [J. Math. Pures Appl., IX. Sér. 12, 1–82 (1933; Zbl 0006.16702)] 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 [Math. Ann., Berlin, 117, 764–775 (1941; JFM 67.0842.02) and Math. Ann. 117, 764–775 (1941; Zbl 0024.13505)] 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 [J. Fac. Sci., Univ. Tokyo, Sect. I 9, 59–102 (1961; Zbl 0111.38502)] and R. Finn [Acta Math. 105, 197–244 (1961; Zbl 0126.42203)]. H. Fujita and H. Morimoto [Sapparo, Japan, 1995. GAKUTO 1995. Int. Ser., Math. Sci. Appl. 10, 53–61 (1997; Zbl 0887.35118)] proved existence theorem for flows, which are close to potential ones). M. V. Korobkov et al. (see e.g., [M. Korobkov et al., J. Math. Pures Appl. (9) 101, No. 3, 257–274 (2014; Zbl 1331.35263)]) 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 apriori estimate of solution under following additional conditions: the flow has a symmetries plane; the flux through the inner domain boundary is positive. This estimate implies solvability of the problem.

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 [J. Fac. Sci., Univ. Tokyo, Sect. I 9, 59–102 (1961; Zbl 0111.38502)] and R. Finn [Acta Math. 105, 197–244 (1961; Zbl 0126.42203)]. H. Fujita and H. Morimoto [Sapparo, Japan, 1995. GAKUTO 1995. Int. Ser., Math. Sci. Appl. 10, 53–61 (1997; Zbl 0887.35118)] proved existence theorem for flows, which are close to potential ones). M. V. Korobkov et al. (see e.g., [M. Korobkov et al., J. Math. Pures Appl. (9) 101, No. 3, 257–274 (2014; Zbl 1331.35263)]) 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 apriori estimate of solution under following additional conditions: the flow has a symmetries plane; the flux through the inner domain boundary is positive. This estimate implies solvability of the problem.

##### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |