Large, global solutions to the Navier-Stokes equations, slowly varying in one direction

In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation.Comment: References adde

On the global wellposedness of the 3-D Navier-Stokes equations with large initial data

We give a condition for the periodic, three dimensional, incompressible Navier-Stokes equations to be globally wellposed. This condition is not a smallness condition on the initial data, as the data is allowed to be arbitrarily large in the scale invariant space $B^{-1}\_{\infty,\infty}$, which contains all the known spaces in which there is a global solution for small data. The smallness condition is rather a nonlinear type condition on the initial data; an explicit example of such initial data is constructed, which is arbitrarily large and yet gives rise to a global, smooth solution

Tempered distributions and Fourier transform on the Heisenberg group

The final goal of the present work is to extend the Fourier transform on the Heisenberg group \H^d, to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on \H^dis no longer a function on \H^d : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable functionto be a mapping on the set~\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}endowed with a suitable distance \wh d.This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on \H^d by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on \H^d that are independent of the vertical variable. We also provide other examples

Sums of large global solutions to the incompressible Navier-Stokes equations

Let G be the (open) set of~$\dot H^{\frac 1 2}$ divergence free vector fields generating a global smooth solution to the three dimensional incompressible Navier-Stokes equations. We prove that any element of G can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of G if the variation is slow enough. This result implies that through any point in G passes an uncountable number of arbitrarily long segments included in G.Comment: Accepted for publication in Journal f\"ur die reine und angewandte Mathemati

Global regularity for some classes of large solutions to the Navier-Stokes equations

In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared'' (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared'' situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation