Stability by rescaled weak convergence for the Navier-Stokes equations

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We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem: if a sequence (u₀, ₙ)ₙ∈ of initial data, bounded in some scaling invariant space, converges weakly to an initial data u₀ which generates a global regular solution, does u₀, ₙ generate a global regular solution? A positive answer in general to this question would imply global regularity for any data, through the following examples u₀,ₙ = n ₀(n·) or u₀,ₙ = ₀(·-xₙ) with |xₙ|→ ∞. 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.