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Abstract
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Fluid flows are ubiquitous in scientific and engineering applications as well as in our daily life. It is well known that the motion of fluid flows is governed by the Navier-Stokes equations. Despite its wide spread applications, the global existence and regularity of the three dimensional incompressible Navier-Stokes equations remains to be one of the most challenging open questions in fluid dynamics and mathematics. This challenging question is one of the Seven Millennium Prize Problems posted by the Clay Mathematics Institute. In this lecture, I will review some classical results as well as the recent developments for the 3D Euler and Navier-Stokes equations, and explain why this problem is so challenging. Finally, we will report some recent progress in searching for potential finite time singularities of the 3D Euler equations and related models using an integrated analysis and computation approach. We will also discuss possible implication of the 3D Euler singularities on the 3D Navier-Stokes equations.
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