On the Morse–Sard theorem for the sharp case of Sobolev mappings and its applications in fluid mechanics

We establish Luzin N- and Morse–Sard properties for the sharp case of Sobolev–Lorentz classes $W^k_p(R^n,R^m)$ under minimal integrability assumptions (that garantee the continuity of a mapping only, i.e., $p=n/k$). Using these results we prove that almost all level sets of such functions are finite disjoint unions of $C^1$–smooth compact manifolds of dimension $n-m$ (despite the fact that a function itself is not $C^1$ — it is continuous only).
These results helped in mathematical fluid mechanics — for the so-called Leray's problem, which remained open for more than 80 years (starting from the publication of the famous paper of Jean Leray 1933 ). Namely, for plane and axially symmetric spatial flows the existence theorem was proved for boundary value problem of stationary Navier-Stokes equations in bounded domains under necessary and sufficient condition of zero total flux.