On the Serrin-Type Condition on One Velocity Component for the Navier-Stokes Equations

초록

In this paper we consider the regularity problem of the Navier-Stokes equations in R-3. We show that the Serrin-type condition imposed on one component of the velocity u(3 )is an element of Lp (0, T; Lq(R-3)) with 2/p + 3/q < 1,3 < q <= + infinity implies the regularity of the weak Leray solution u : R(3 )x (0,T) -> R-3, with the initial data belonging to L-2 (R-3) boolean AND L-3 (R-3). The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.