Some advances on the geometric non blow-up criteria of incompressible flows

Léo Agélas (1)

1 IFPEN - IFP Energies nouvelles

One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. Recently, from a numerical point of view, Luo & Hou presented a class of potentially singular solutions to the Euler equations in a fluid with solid boundary [70, 71]. Furthermore, in two recent papers [85, 86], Tao indicates a significant barrier to establishing global regularity for the three-dimensional Euler and Navier-Stokes equations, in that any method for achieving this must use the finer geometric structure of these equations. In this paper, we show that the singularity discovered by Luo & Hou which lies right on the boundary is not relevant in the case of the whole domain ℝ^3. We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations and not shared by the averaged Navier-Stokes and generalised Euler equations introduced respectively in [85, 86], is the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations which characterize their special geometric structures allowed to establish a new geometric non blow-up criterion and to improve greatly the Beale-Kato-Madja regularity criterion.

Mots clés: 3D Euler equations; 3D Navier-Stokes equations; 2D Quasi-Geostrophic equation; Finite time singularities; Geometric properties for non blow-up

Domaines: Equations aux dérivées partielles [math.AP]; Dynamique des Fluides [physics.flu-dyn]

Léo Agélas. Some advances on the geometric non blow-up criteria of incompressible flows. 2017. ⟨hal-01380349v3⟩