Leray's problem on existence of steady-state solutions for the navier-stokes flow

This is a survey of results on the Leray problem (1933) for the nonhomogeneous boundary value problem for the steady Navier-Stokes equations in a bounded domain with multiple boundary components. The boundary conditions are assumed only to satisfy the necessary requirement of zero total flux. The authors have proved that the problem is solvable in arbitrary bounded planar or threedimensional 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. Similar existence results (without any restrictions on fluxes) are proved for steady Navier-Stokes system in two- and three-dimensional exterior domains with multiply connected boundary under assumptions of axial symmetry. In particular, it was shown that in domains with two axes of symmetry and for symmetric boundary datum, the two-dimensional exterior problem has a symmetric solution vanishing at infinity.