University of Nebraska--Lincoln
Date & Time
May 7, 2021, 2:00 pm – 3:00 pm
Beauty and the Beast: The Flame Equation in 1D and 2D
Abstract:The Kuramoto-Sivashinsky equation (KSE) is a beautiful and highly chaotic dynamical system that arises in flame fronts, plasmas, crystal growth, and many other phenomena. Due to its lack of a maximum principle and its advective-type nonlinearity, the KSE is often studied as an analogue to the 3D Navier-Stokes equations. Much progress has been made on the beautiful 1D KSE since roughly 1984, but for the beastly 2D KSE, even global well-posedness remains a major open question. Moreover, as has been demonstrated recently by Kostianko, Titi, and Zelik, standard regularizations that work well for Navier-Stokes fail when applied to even the 1D KSE. Despite this, we present linear modifications of the 2D KSE which allow for global well-posedness, while still retaining many horrifying features of the 2D KSE. We will also discuss some recent results on Prodi-Serrin type results for the 2D KSE. This talk will describe key ideas of the analysis and show colorful movies of solutions. It should be accessible to students as well.