Compressible euler and kuramoto-sivashinsky-type equations

可压缩欧拉和 kuramoto-sivashinsky 型方程

• 批准号: 341834-2007
• 负责人: Fetecau, Razvan
• 金额: $ 0.95万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363342
• 关键词: Compressible; euler; kuramoto; sivashinsky; type

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.

我目前的研究兴趣在于非线性偏微分方程的一般领域，涵盖了从严格分析到渐近和数值方法的广泛范围。特别是，我对可压缩欧拉方程和Kuramoto-Sivashinsky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类方程唯一熵解的整体存在性。这是一个众所周知的问题，在过去50年中引起了极大的兴趣和努力。我想要追求的主要思想是使用一种正则化，以前没有用于可压缩流体。Leray（1934）以前曾用这种正则化方法来证明三维不可压Navier-Stokes方程整体弱解的存在性。Bhat（哥伦比亚）和我已经将这些想法应用于Burgers方程，结果非常令人鼓舞。我研究计划的另一个方向是致力于改进Kuramoto-Sivashinsky（KS）和其他相关方程的现有能量边界。导出KS方程的精确能量边界也是一个非常困难的问题。在过去的二十年里，只有少数论文对这一问题做出了贡献，并且从数值实验中观察到，结果与被认为是最好的边界相差甚远。我最近对这个方程产生了兴趣，我想继续这个研究方向，也许在开始时有更适度的期望，当我想专注于改进文献中提出的各种修改的KS方程的界限，并解决多维情况。