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

Agélas, Léo

doi:10.1016/j.anihpc.2019.04.003

Agélas, Léo

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1503-1537.

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Résumé

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 $R^{3}$ . 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.

DOI : 10.1016/j.anihpc.2019.04.003

Classification : 35Q30, 35Q31, 76B60, 76B65, 76B03
Mots-clés : 3D Euler equations, 3D Navier-Stokes equations, 2D Quasi-Geostrophic equation, Finite time singularities, Geometric properties for non blow-up

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Agélas, Léo. A new path to the non blow-up of incompressible flows. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 6, pp. 1503-1537. doi : 10.1016/j.anihpc.2019.04.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2019.04.003/

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