Compressible euler and kuramoto-sivashinsky-type equations

我目前的研究兴趣在于非线性偏微分方程组的一般领域，涵盖了从严格分析到渐近性和数值计算的广泛范围。我尤其对可压缩欧拉方程和Kuramoto-Sivashinsky类型方程感兴趣。关于可压缩欧拉方程，我主要想从新的角度研究这些方程的唯一熵解的整体存在性。这是一个臭名昭著的问题，在过去的50年里吸引了人们的极大兴趣和努力。我想要追求的主要思想是使用一种以前没有用于可压缩流体的正则化。这种正则化方法曾被Leray(1934)用来得出$3D$不可压缩的Navier-Stokes方程弱解的整体存在性。我和我的合作者H.Bhat(Columbia)已经将这些想法应用到Burgers方程中，结果非常令人鼓舞。我的研究计划的另一个方向是致力于改善Kuramoto-Sivashinsky(KS)和其他相关方程的现有能界。得到KS方程的能量界也是一个非常困难的问题。在过去的二十年里，只有少数几篇论文对这一主题做出了贡献，而且从数值实验中观察到的结果与人们认为的最佳界限相去甚远。我最近对这个方程很感兴趣，我想继续这个研究方向，也许一开始就有更低的期望，那时我想专注于改善文献中提出的各种新修改的KS方程的界限，并处理多维情况。

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.