Applied Mathematics Colloquium: Dr. Vincent Martinez
Friday, December 7, 2018 · 2 PM - 3 PM
Title: Asymptotic coupling and uniqueness of invariant measures for the damped, stochastically-driven Korteweg de Vries equation
Speaker: Vincent Martinez, Hunter College
Abstract: In their 1967 seminal paper, Foias and Prodi captured precisely a notion of finitely many degrees of freedom in the context of the two-dimensional (2D) incompressible Navier-Stokes equations (NSE). In particular, they proved that if a sufficiently large spectral projection of the difference of two solutions converge to 0 asymptotically in time, then the corresponding complementary projection of their difference must also converge to 0 in the infinite-time limit. In other words, the high modes are ``eventually enslaved`` by the low modes. One could thus define the number of degrees of freedom of the flow to be the smallest number of modes needed to guarantee this convergence for a given flow, insofar as it is represented by a solution to the NSE. This property has since led to several developments in the understanding and application of the long-time behavior of solutions to the NSE, for instance, in the context of turbulence, data assimilation, and the existence of determining forms. In this talk, we will discuss this asymptotic enslavement property as it regards the issue of uniqueness of invariant measures for stochastically forced equations, particularly the damped, stochastic KdV equation.