No blow-up of the 3D incompressible Euler equations

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,+∞[;W r,q(R3))3 with q> 1, r> 1+ 3 q, of the Euler equations as soon as the initial data u0 ∈ W r,q(R3)3. This result gives a positive answer to the open problem about existence and smoothness of solutions of Euler equations.