Compressible euler and kuramoto-sivashinsky-type equations

点击翻译按钮获取中文摘要

My current research interests lie in the general area of nonlinear partial differential equations, covering a broad spectrum of methods ranging from rigorous analysis to asymptotics and  numerics.  In particular, I have interests in the compressible Euler  and the Kuramoto-Sivashinsky-type equations.Regarding the compressible Euler equations, I would mainly like to investigate from new viewpoints the global existence of unique entropy solutions for these equations. This is a notorious problem and has attracted a great deal of interest and effort during the last 50 years. The main idea that I want to pursue is to use a type of regularization that hasn't been used before for compressible fluids. This type of regularization was previously used by Leray (1934) to conclude global existence of weak solutions of the $3D$ incompressible Navier-Stokes equations.  My collaborator, H. Bhat (Columbia), and I already applied these ideas to the Burgers equation and the results are very encouraging.The other direction of my research program is to work on improving existing energy bounds for the Kuramoto-Sivashinsky (KS) and other related equations. The issue of deriving sharp energy bounds for the KS equation is also a very hard problem. There is only a handful of papers that made a contribution to the subject over the last two decades and the results are quite far from what is believed to be the best bound, as observed from numerical experiments. I have very recently taken an interest in this equation and I want to pursue this research direction, with perhaps more modest expectations at the beginning, when I want to focus on improving bounds for the various  modified KS equations proposed in the literature and also tackle the multidimensional case.