The generalized Korteweg-de Vries equation (gKdV) is obtained by changing the nonlinearity of the famous (quadratic) Korteweg-de Vries equation. We consider the quintic power nonlinearity, for which the gKdV equation is critical for the L^2-norm.
It was established in the early 2000s that for initial data with sufficiently large mass, solutions can blow up in finite time through the concentration (or “bubbling”) of a solitary wave. In all known constructions, the spatial location of this solitary wave goes to infinity as time approaches the blow-up time.
In the first part of this talk, I will review these classical blow-up results. In the second part, I will present new results obtained in collaboration with Yvan Martel, in which we construct solutions that blow up in finite time at a finite spatial point. These new constructions also yield blow-up rates that are closer to the self-similar regime. They are obtained through the interaction between a solitary wave and a cusp-shaped function.

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 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, 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.