Abstract
We 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.
Highlights
Abstract. — We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system
The goal of this work is to try to understand if such a property, which we can call “weak stability”, holds more generally: we would like to address the question of weak stability: If (u0,n)n∈N, bounded in some scale invariant space X0, converges to u0 in the sense of distributions, with u0 giving rise to a global smooth solution, is it the case for u0,n when n is large enough?
We look for the global solution associated with u0,n under the form un = uanp,εp + Rn,ε with uanp,εp d=ef
Summary
The results recalled above tend to suggest that the initial data should satisfy some sort of smallness assumption if one is to prove global existence and uniqueness of solutions. The goal of this work is to try to understand if such a property, which we can call “weak stability”, holds more generally: we would like to address the question of weak stability: If (u0,n)n∈N, bounded in some scale invariant space X0, converges to u0 in the sense of distributions, with u0 giving rise to a global smooth solution, is it the case for u0,n when n is large enough?. — Let us recall that it is proved in [1] (see [21] for the case of Besov spaces Bp−,q1+3/p) that the set of initial data generating a global solution is open in BMO−1. The question asked above addresses the case when the sequence converges non longer strongly, but in the sense of distributions
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