Some advances on the geometric non blow-up criteria of incompressible flows
Résumé
One of the most challenging questions in fluid dynamics is whether the three-dimensional incompressible Navier-Stokes, Euler and two-dimensional Quasi-Geostrophic (2D QG) equations can develop a finite-time singularity from smooth initial data. The development of finite-time singularities was first investigated via energy methods which focus on global features or on point-wise Eulerian features of the flow. This comes at the disadvantage of neglecting the structures and physical mechanisms of the flow evolution. A strategy to overcome such shortcomings was established by focusing more on geometrical properties and flow structures, such as vortex tubes or vortex lines. However, all these recent geometric non blow-up criteria use the Lagrangian formulation of Incom-pressible Inviscid Flows, which requires much more computational effort than an Eulerian framework. In this paper, we improve these geometric non blow-up criteria while keeping an Eulerian setting by using Pshenichnyi's achievements in its investigation of necessary extremum conditions. By using also a dyadic unity partition of Littlewood decomposition, we established non blow-up criteria involving the dyadic BMO spaces known to be suitable for odd functions [86]. We thus obtained new insights on the growth of the maximum vorticity. Another highlight of our analysis is it allows to study the Navier-Stokes, Euler and 2D QG equations in a same framework. Our Eulerian geometric non blow-up criteria should give also new impetus to the numerical experiments due to their ease of implementation in comparison with Lagrangian geometric non blow-up criteria.
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