Mathematics Dept.
  Events
Juncheol Pyo
Pusan National University and UBC
Tue 2 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105
Solitons for the mean curvature flow and inverse mean curvature flow
MATH 105
Tue 2 Oct 2018, 3:30pm-4:30pm

Abstract

 Self-similar solutions and translating solitons are not only special solutions of mean curvature flow (MCF) but a key role in the study of singularities of MCF. They have received a lot of attention. We introduce some examples of self-similar solutions and translating solitons for the mean curvature flow (MCF) and give rigidity results of some of them. We also investigate self-similar solutions and translating solitons to the inverse mean curvature flow (IMCF) in Euclidean space. 
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Christos Mantoulidis
MIT
Tue 9 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105
Minimal surfaces and the Allen-Cahn equation on 3 manifolds
MATH 105
Tue 9 Oct 2018, 3:30pm-4:30pm

Abstract

   The Allen--Cahn equation is a semi-linear PDE that produces minimal surfaces via a certain singular limit. We will describe recent work proving index, multiplicity, and curvature estimates in the context of an Allen--Cahn min-max construction in a 3-manifold. Our results imply, for example, that in a 3-manifold with a generic metric, for every positive integer p, there is an embedded two-sided minimal surface of Morse index p. This is joint with Otis Chodosh.
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Hyunju Kwon
UBC
Tue 23 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105
Global Navier-Stokes flows for non-decaying initial data with slowly decaying oscillation
MATH 105
Tue 23 Oct 2018, 3:30pm-4:30pm

Abstract

We consider the Cauchy problem of 3D incompressible Navier-Stokes equations for uniformly locally square integrable initial data. The existence of a time-global weak solution has been known, when the square integral of the initial datum on a ball vanishes as the ball goes to infinity. For non-decaying data, however, the only known global solutions are either for perturbations of constants or when the velocity gradients are in Lp with finite p. In this talk, I will outline how to construct global weak solutions for general non-decaying initial data whose local oscillations slowly decay.
This is a joint work with Tai-Peng Tsai.
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