Events for 09/19/2017 from all calendars

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: James Kelliher, UC Riverside

Title: Nonlinear PDEs Seminar

Abstract:

Title: Bounded Vorticity, unbounded velocity solutions to the 2D Euler equations.

Abstract: The pioneering work on bounded vorticity solutions to the 2D Euler equations was done by Yudovich in the early 1960s, working in a bounded domain. He proved existence and uniqueness of such solutions. Subsequently, existence was shown to hold in much weaker settings (vorticity lying in any Lebesgue space), but uniqueness has only ever been extended to an incrementally larger class of initial data.

Yudovich's theory extends easily to the full plane (indeed it is slightly less technical there) as long as the velocity is assumed to decay sufficiently rapidly at infinity that the Biot-Savart law holds. In 1995, Ph. Serfati established the existence and uniqueness of bounded vorticity solutions having no decay at infinity. Such solutions very much violate the Biot-Savart law, but Serfati discovered an identity that the solutions hold that can be used as a kind of substitute for that law.

The boundedness of the velocity was very important in Serfati's argument, yet there is room in his identity to accommodate some growth of the velocity at infinity. I will speak on ongoing joint work with Elaine Cozzi in which we exploit Serfati's identity to obtain existence and uniqueness classes allowing growth at infinity as large as possible (without assuming any special symmetry of the initial data). Roughly speaking, we show that existence can be achieved only for very slowly growing velocities, but that uniqueness holds for velocities growing slower than the square root of the distance from the origin. We also consider the issue of continuous dependence on initial data, which is already an interesting problem even in Yudovich's original setting.

Analysis/PDE Reading Seminar

Time: 4:00PM - 5:00PM

Location: BLOC 624

Speaker: Eviatar Procaccia, TAMU

Title: Introduction to SLE (Schramm-Lowner Evolution)

Abstract: SLE is a family of random curves, which is proved to be the scaling limit of many discrete random interfaces, common in statistical physics. The study of SLE gave birth to beautiful techniques such as Smirnov's discrete holomorphic functions, and study of fractal dimension of random curves. In this first talk I will introduce the subject of SLE, starting from a background of models in statistical physics (percolation, loop erased radnom walk, etc), and then continuing with the fantastic discoveries of Oded Schramm, Greg Lawler, Wendelin Werner, Steffen Rhode and others.

Student/Postdoc Working Geometry Seminar

Time: 5:00PM - 6:00PM

Location: BLOC 628

Speaker: A. Conner, TAMU

Title: Computer calculations with tensors