In this talk, we will discuss recent advances towards understanding the regularity hypotheses in the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations. We show that, in general, their theorem cannot be extended to any Sobolev space on the 1D torus. This is demonstrated by constructing arbitrarily small solutions with a sequence of nonlinear oscillations, known as plasma echoes, which damp at a rate arbitrarily slow compared to the linearized Vlasov equations. Some connections with hydrodynamic stability problems will be discussed if time permits.
Jacob earned his PhD in 2011 from UCLA under advisors Andrea Bertozzi and Joey Teran. He went on to the Courant Institute as an NSF post-doc and in 2014 he joined the faculty at the University of Maryland, College Park. In 2015 he received a Sloan research fellowship and in 2016 he was awarded an NSF CAREER grant. Most of his research is focused on hydrodynamic stability at high Reynolds number and collisionless kinetic theory, but has also included the analysis of Keller-Segel models from mathematical biology, calculus of variations, and scientific computing.