turkmath.org

In this talk, I 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, I will show the universality of the critical 1/3 Hölder 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, I 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.