Some remarks about the possible blow-up for the Navier-Stokes equations. Issue 12 (2nd December 2019)

• Record Type: Journal Article
• Authors: Chemin, Jean-Yves; Gallagher, Isabelle; Zhang, Ping
• Is Part Of: Communications in partial differential equations. Volume 44:Issue 12(2019)
• Page Start: 1387
• Page End: 1405
• Publication Date: 2019-12-02
• Subjects: Anisotropic Littlewood-Paley Theory -- blow-up criteria -- incompressible Navier-Stokes equations; 35Q30 -- 76D03; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2019.1641725
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: In this work, we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant norms. In particular we prove that it is not possible for one component of the velocity field to tend to 0 too fast near blow up. We also introduce a space "almost" invariant under the action of the scaling such that if one component of the velocity field measured in this space remains small enough, then there is no blow up.