Direction of Vorticity and Regularity up to the Boundary: On the Lipschitz-Continuous Case

Abstract

In their famous 1993 paper, Constantin and Fefferman consider the evolution Navier-Stokes equations in the whole space R 3 and prove, essentially, that if the direction of the vorticity is Lipschitz continuous in the space variables, during a given time-interval, then the corresponding solution is regular. Since Lipschitz-continuity is a very natural, basic, property, it looks interesting to go further in this particular direction. In this paper, we consider the initial-boundary value problem for the Navier-Stokes equations in a regular, bounded, domain under a slip boundary condition, and prove regularity of the solution, up to the boundary, under a weakened Lipschitz-continuity assumption on the direction of the vorticity. The interest of our result highly relies on the fact that the Lipschitz-continuity coefficient g(x, t )i ssharp. This means, in a sense, that our finding possesses the same level of accuracy as that of the classical "Prodi-Serrin" type conditions; see the introductory section. It should be remarked that a similar result was already obtained in the 2009 paper by Beirao da Veiga and Berselli. In the latter, the proof of an analogous sharp result was shown under the assumption of 1 -H˝ older continuity on the direction of vorticity. The authors also claimed, correctly, that by the same ideas the proof of such a result could be extended to H˝ older exponents β ∈ )0 , 1 ). However the proofs would be extremely involved. On the contrary, the proof followed in this paper treat the Lipschitz case is definitely more elementary than any other proof, even if restricted to the whole space case.