The flux problem for the Navier-Stokes equations

This is a survey of results on the Leray problem (1933) for the Navier-Stokes equations of an incompressible fluid in a domain with multiple boundary components. Imposed on the boundary of the domain are inhomogeneous boundary conditions which satisfy the necessary requirement of zero total flux. The authors have proved that the problem is solvable in arbitrary bounded planar or axially symmetric domains. The proof uses Bernoulli's law for weak solutions of the Euler equations and a generalization of the Morse-Sard theorem for functions in Sobolev spaces. New a priori bounds for the Dirichlet integral of the velocity vector field in symmetric flows, as well as estimates for the regular component of the velocity in flows with singularities of source/sink type are presented.