Abstract: 

In 1993, Constantin and Fefferman established an interesting connection between vorticity alignment and the regularity of Navier-Stokes solutions: they demonstrated that spatial Lipschitz continuity of vorticity directions precludes singularity formation. Extending this paradigm, we investigate a scale-invariant scenario where vorticity remains confined within a double cone. We prove that such solutions stay regular - in other words, near any potential singularity, the vorticity direction set has to touch every great circle on the unit sphere.The proof is inspired by the Kelvin-Helmholz law for ideal fluids and the critical regularity theory for axisymmetric solutions. Quantitative intervals of regularity also plays an important role in the analysis. Based on joint work with Zhen Lei and Gang Tian.

Time:

April 7, 2025, 4pm-5pm