We establish Luzin N and Morse-Sard properties for BV 2 -functions defined on open domains in the plane. Using these results we prove that almost all level sets are finite disjoint unions of Lipschitz arcs whose tangent vectors are of bounded variation. In the case of W 2,1 -functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of C 1 -arcs whose tangent vectors are absolutely continuous.

We study the nonhomogeneous boundary value problem for the Navier-Stokes equations of steady motion of a viscous incompressible fluid in arbitrary bounded multiply connected plane or axiallysymmetric spatial domains. We prove that this problem has a solution under the sole necessary condition of zero total flux through the boundary. The problem was formulated by Jean Leray 80 years ago. The proof of the main result uses Bernoulli's law for a weak solution to the Euler equations.

We study the nonhomogeneous boundary value problem for Navier-Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional bounded multiply connected domain Ω = Ω 1 \ Ω 2 , Ω 2 ⊂ Ω 1 . We prove that this problem has a solution if the flux F of the boundary datum through ∂Ω 2 is nonnegative (

For the incompressible Navier-Stokes equations in the 3D half space, we show the existence of forward self-similar solutions for arbitrarily large self-similar initial data.