This 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.
