Partial Regularity and the Navier-Stokes Equations

Abstract

A sufficiently regular strong solution is known to be unique in the class of Leray–Hopf weak solutions. However, weak solutions are not known to be unique. The fundamental regularity problem for the Navier–Stokes equations in three space dimensions remains open: even if f = 0, one does not know whether weak solutions remain smooth for all time. A suitable weak solution can be singular only on a rather small set. Results of this type, called “partial regularity theorems,” are well known in the theory of minimal surfaces and quasi-linear elliptic systems. It was Scheffer's remarkable idea to study the Navier–Stokes equations from this point of view.