PDE and Applied Mathematics Seminar

Spring 2014

Organized by Dmitry Golovaty and Peter Gordon

February 4, (CAS 220D, 4:30-5:30pm) Dmitry Ryabogin (Kent State University) will speak on "Several problems of uniqueness of convex bodies".

Abstract: Assume that we are given two convex bodies $K$ and $L$ in ${\mathbb R^n}$, $n\ge 2$, such that the corresponding projections (onto every hyperplane passing through the origin) are congruent (there is a translation and a rotation bringing one projection into another). Do the bodies $K$ and $L$ coincide (up to reflection and translation)? We will discuss this open question and other related problems of convex geometry. This talk should be clear for graduate students.

March 13, (CAS 220D, 4:30-5:30 pm) Fedor Nazarov (Kent State University) will give "Two remarks on the dissipative surface quasi-geostrophic equation".

Abstract: We discuss the results concerning the well-posedness of the 2D dissipative quasi-geostrophic equation in the critical regime (Laplacian to the power 1/2) and the eventual regularity in the supercritical (any small positive power of the Laplacian) regime. I do not expect to be able to present much more than just rough outlines of the main ideas of the proofs this time, but should enough interest arise, I'll be happy to give a series of lectures with full details. The presentation is based on the papers "Global well-posedness for the critical 2D dissipative quasi-geostrophic equation" by A. Kiselev, F. Nazarov, A. Volberg and "Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation" by M. Dabkowski (both available on arXiv).

April 8, (CAS 220D, 3:30-4:30pm) Truyen Nguyen (University of Akron) will speak on "Gradient estimates and global existence of smooth solutions to a cross-diffusion system. "

Abstract: We study the initial value problem for a cross-diffusion system in bounded domains of any dimension. This system was proposed by Shigesada, Kawasaki and Teramoto in 1979 to describe the habitat segregation phenomena between two species which are competing in the same domain. A global-time existence of smooth solutions is established by deriving global $W^{1,p}$-estimates for weak solutions to a class of nonlinear parabolic equations with nonlinear diffusion. A perturbation argument and a new double scaling technique are introduced to obtain the estimates for large reaction terms. This is joint work with Luan Hoang and Tuoc Phan.

April 15, 17, 24, 29 and May 1, (Applied Math. Lab., 3:30-5:00 pm), Mini-course: Well-posedness of dissipative surface quasi-geostrophic equation by Fedor Nazarov (Kent State).

April 22, (CAS 220D, 2:30-3:30pm) Irene Fonseca (Carnegie Mellon University) will speak on "Variational Methods for Crystal Surface Instability."

Abstract: Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of a material void in a linearly elastic solid may be deduced from smoothness and convexity properties of the interfacial energy. In addition, short time existence, uniqueness, and regularity for an anisotropic surface diffusion evolution equation with curvature regularization are proved in the context of epitaxially strained two-dimensional films. This is achieved by using the $H^{-1}$-gradient flow structure of the evolution law, via De Giorgi's minimizing movements. This seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity.

Previous seminars:
Fall 2013
Spring 2013
Fall 2012