Improved Geometric Conditions for Non-Blowup of the 3D Incompressible Euler Equation

Abstract

This is a follow-up of our recent article Deng et al. (2004 Deng, J.,Hou, T. Y., Yu, X. (2004). ). In Deng et al. (2004), we derive some local geometric conditions on vortex filaments which can prevent finite time blowup of the 3D incompressible Euler equation. In this article, we derive improved geometric conditions which can be applied to the scenario when velocity blows up at the same time as vorticity and the rate of blowup of velocity is proportional to the square root of vorticity. This scenario is in some sense the worst possible blow-up scenario for velocity field due to Kelvin's circulation theorem. The improved conditions can be checked by numerical computations. This provides a sharper local geometric constraint on the finite time blowup of the 3D incompressible Euler equation.

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