| One of the most challenging questions in fluid dynamics is whether the incompressible Euler equations can develop a finite-time singularity from smooth initial data. In this paper, we found that local geometry regularity of vorticity leads to a strong dynamic depletion of the nonlinear vortex stretching, thus avoiding finite-time singularity formation. Then, we prove the existence and uniqueness of global strong solutions in $C([0,+\infty[;W^{r,q}(\mathbb{R}^3))^3$ with $q>1$, $r>1+\frac{3}{q}$, of the Euler equations as soon as the initial data $u_0\in W^{r,q}(\mathbb{R}^3)^3$. This result gives a positive answer to the open problem about existence and smoothness of solutions of Euler equations. |
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