Global regularity to the Navier-Stokes equations for a class of large initial data

Abstract

In [5], Chemin, Gallagher and Paicu proved the global regularity of solutions to the classical Navier-Stokes equations with a class of large initial data on T2 × R. This data varies slowly in vertical variable and has a norm which blows up as the small parameter ( represented by ǫ in the paper) tends to zero. However, to the best of our knowledge, the result is still unclear for the whole spaces R3. In this paper, we consider the generalized Navier-Stokes equations on Rn(n ≥ 3): ∂tu + u · ∇u + Dsu + ∇P = 0, div u = 0. For some suitable number s, we prove that the Cauchy problem with initial data of the form u0ǫ(x) = (v0h(xǫ), ǫ−1v0n(xǫ))T , xǫ = (xh, ǫxn)T , is globally well-posed for all small ǫ > 0, provided that the initial velocity profile v0 is analytic in xn and certain norm of v0 is sufficiently small but independent of ǫ. In particular, our result is true for the n-dimensional classical Navier-Stokes equations with n ≥ 4 and the fractional Navier-Stokes equations with 1 ≤ s < 2 in 3D.

Keywords

• large data
• global solution
• slowly varying