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

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

Wellposedness and stability results for the Navier-Stokes equations in R3{\mathbf R}^{3}

In a previous work, we presented a class of initial data to the three dimensional, periodic, incompressible Navier-Stokes equations, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is twofold. First, we adapt the construction to the case of the whole space: we prove that if a certain nonlinear function of the initial data is small enough, in a Koch-Tataru type space, then there is a global solution to the Navier-Stokes equations. We provide an example of initial data satisfying that nonlinear smallness condition, but whose norm is arbitrarily large in $C^{-1}$. Then we prove a stability result on the nonlinear smallness assumption. More precisely we show that the new smallness assumption also holds for linear superpositions of translated and dilated iterates of the initial data, in the spirit of a construction by the authors and H. Bahouri, thus generating a large number of different examples.Comment: 28 pages misprints correcte

ON LOWER BOUNDS AT BLOW UP OF SCALE INVARIANT NORMS FOR THE NAVIER-STOKES EQUATIONS

In this work we investigate the problem of preventing the incompressible 3D Navier-Stokes from developing singularities with the control of one component of the velocity field only in L ∞ norm in times with values in a scaling invariant space. We introduce a space " almost " invariant under the action of the scaling such that if one component measured in this space remains small enough, then there is no blow up

ON THE STABILITY OF GLOBAL SOLUTIONS TO THE THREE DIMENSIONAL NAVIER-STOKES EQUATIONS

International audienceWe prove a weak stability result for the three-dimensional homogeneous incom-pressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence (u_{0,n})_{ n∈\in N} of initial data, bounded in some scaling invariant space, converges weakly to an initial data u0 which generates a global smooth solution, does u0,n generate a global smooth solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples u_{0,n} = nϕ0(n·) or u_{0,n}) = ϕ0(· − x_n) with |x_n| → ∞. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations