U.S.-U.K. Doctoral Dissertation Enhancement Project: The Many Aspects of Fluids

Oise 06-30623 Schonbek，Maria这是一个支持英国牛津大学博士论文增强研究的项目。美国首席研究员兼顾问是加州大学圣克鲁斯分校的Maria Schonbek。她的学生是Clayton Bjorland，她的英国合作者是Endre Suli。在这个项目中，Clayton Bjorland将考虑与一组非线性微分方程组的存在性和渐近性有关的问题：粘性Camassa-Holm方程(VCHE)，也称为Navier-Stokes-Alpha方程(NS-Alpha)。这些方程源于对浅水方程的研究。它们也被导出为一个“过滤”的Navier-Stokes方程，它遵循修正的Kelvin环流定理。Navier-Stokes-Alpha方程与著名的Navier-Stokes方程密切相关，但过滤器允许获得目前无法获得的Navier-Stokes方程的边界，从而使NS-Alpha方程在某种意义上更适合于计算湍流。人们希望这些方程可以很好地近似于流体方程的主要模型--纳维斯托克斯方程。克莱研究所提出的千禧年问题之一是关于纳维斯托克斯方程解的正则性，这是多年来一直无法解决的重要经典问题之一。纳维-斯托克斯方程描述了液体和气体的运动。虽然它们是在19世纪被发现的，但人们对它们的理解仍然不是很好。问题是要在数学理论上取得进展，让我们深入了解这些方程式。

OISE 06-30623Schonbek, MariaThis is a project that supports doctoral dissertation enhancement research at Oxford University in the U.K. The U.S. principal investigator and advisor is Maria Schonbek from the University of California, Santa Cruz. Her student is Clayton Bjorland and her U.K. collaborator is Endre Suli. In this project Clayton Bjorland will consider questions related to the existence and asymptotic behavior of a system of nonlinear differential equations: the Viscous Camassa-Holm equations (VCHE), which are also known as Navier-Stokes-alpha equations (NS-alpha). These equations arose from work on shallow water equations. They were also derived as a "filtered" Navier-Stokes equation, which obeys a modified Kelvin circulation theorem along filtered velocities. The Navier-Stokes-alpha equations are closely related the famous Navier-Stokes equations, but the filter allows bounds that are currently unobtainable for the Navier-Stokes equations, and thus makes the NS-alpha equations in some sense better suited for computational turbulence. The hope is that these equations might give good approximations to the Navier-Stokes equations, which are the main model for fluid equations.One of the millennium problems, important classic questions that have resisted solution over the years, from the Clay Institute is in regards to regularity for solutions of the Navier-Stokes equations. The Navier-Stokes equations describe the movement of liquids and gases. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give us insight into these equations.