Recent Advances Concerning the Navier-Stokes and Euler Equations

In this talk we will discuss some recent progress concerning the Navier-Stokes and Euler equations of incompressible fluid. In particular, issues concerning the lack of uniqueness using the convex integration machinery and their physical relevance. Moreover, we will show the universality of the critical 1/3 Holder exponent, conjectured by Onsager for the preservation of energy in Euler equations, by extending the Onsager conjecture for the preservation of generalized entropy in general conservation laws. In addition, we will present a blow-up criterion for the 3D Euler equations based on a class of inviscid regularization for these equations and the effect of physical boundaries on the potential formation of singularity.