Global Regularity to the Navier-Stokes Equations for a Class of Large Initial Data.

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 ∈ 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 • Δu + Dsu + ΔP = 0, div u = 0. For some suitable number s, we prove that the Cauchy problem with initial data of the form u0∈(x) = (v0h(x∈), ∈1v0n (x∈))T , x∈ = (xh, ∈xn)T , is globally well-posed for all small ∈ > 0, provided that the initial velocity profile v0 is analytic in x0 and certain norm of v0 is suffciently small but independent of ∈. 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. [ABSTRACT FROM AUTHOR]