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Applied Mathematics Colloquium: Dr Adam Larios

University of Nebraska--Lincoln



Date & Time

May 7, 2021, 2:00 pm3:00 pm


Beauty and the Beast: The Flame Equation in 1D and 2D

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.