A new path to the non blow-up of incompressible flows

  • Léo Agélas

    Department of Mathematics, IFP Energies nouvelles, 1-4, avenue de Bois-Préau, F-92852 Rueil-Malmaison, France
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One of the most challenging questions in fluid dynamics is whether the three-dimensional (3D) incompressible Navier-Stokes, 3D 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 [1,2]. Furthermore, in two recent papers [3,4], Tao indicates a significant barrier to establishing global regularity for the 3D 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 . We reveal also that the translation and rotation invariance present in the Euler, Navier-Stokes and 2D QG equations are the key for the non blow-up in finite time of the solutions. The translation and rotation invariance of these equations combined with the anisotropic structure of regions of high vorticity allowed to establish a new geometric non blow-up criterion which yield us to the non blow-up of the solutions in all the Kerr's numerical experiments and to show that the potential mechanism of blow-up introduced in [5] cannot lead to the blow-up in finite time of solutions of Euler equations.

Cite this article

Léo Agélas, A new path to the non blow-up of incompressible flows. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 6, pp. 1503–1537

DOI 10.1016/J.ANIHPC.2019.04.003