摘要：The Euler and Navier–Stokes equations, which describe the motion of inviscid and viscous fluids respectively, have for centuries posed one of the most intriguing open problems in mathematics: can   singularities develop in finite time from smooth initial data? This   question, highlighted by the Clay Mathematics Institute as one of the seven Millennium Prize Problems, lies at the heart of our understanding of fluid motion and turbulence.

     In this talk, I will begin with an overview of these fundamental equations and the role of singularities in fluid dynamics. I will then discuss recent progress and key contributions that shed new light on this long-standing mathematical challenge, focusing on the mechanisms of singularity formation through vortices and droplets.