PDE Seminar Calendar

The PDE seminar is held on Fridays at 3:00 in Kassar 105 unless otherwise specified (*). Coffee and cookies are usually served after the talks.

In Spring 2012, the seminar is co-organized by Walter Strauss and Hongjie Dong.

For more information, or if you would like to give a talk, please send email to Hongjie Dong at hdong_at_dam_dot_brown_dot_edu.

Spring 2012

January
27	Gerard-Varet David, Paris 7 Homogenization and Boundary Layer I will present a joint work with Nader Masmoudi, on the homogenization of elliptic systems with Dirichlet boundary condition, when the coefficients of both the system and the boundary datum are \epsilon-periodic. We show that, as \epsilon\to 0, the solutions converge in L2 with a power rate in \epsilon, and identify the homogenized limit system. Due to a boundary layer phenomenon, this homogenized system depends in a non trivial way on the boundary. Our analysis answers a longstanding open problem, raised by Bensoussan, Lions and Papanicolaou.
February
3	Jared Speck, MIT The Global Stability of the Minkowski Spacetime Solution to the Einstein-Nonlinear Electromagnetic System in Wave Coordinates, (special time: 2:00-2:50 pm) The Einstein-nonlinear electromagnetic system is a coupling of the Einstein field equations of general relativity to a model of nonlinear electromagnetic fields. In this talk, I will discuss the family of covariant electromagnetic models that satisfy the following criteria: i) they are derivable from a sufficiently regular Lagrangian, ii) they reduce to the linear Maxwell model in the weak-field limit, and iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. I will mention several specific electromagnetic models that are of interest to researchers working in the foundations of physics and in string theory. I will then discuss my main result, which is a proof of the global nonlinear stability of the 1 + 3--dimensional Minkowski spacetime solution to the coupled system. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in a wave coordinate gauge. The analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows one to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. The analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor that I develop. Throughout the analysis, I work directly with the electromagnetic fields, thus avoiding the introduction of electromagnetic potentials.
February
3	Mahir Hadzic, MIT The classical Stefan problem and the vanishing surface tension limit We develop a new unified framework for the treatment of well-posedness for the Stefan problem with and without surface tension. This approach yields new estimates for the regularity of the moving surface in the absence of surface tension, which allows us to prove that solutions of the Stefan problem with positive surface tension converge to solutions of the Stefan problem without surface tension. Our techniques rely on a fluid-mechanics inspired approach which, in a suitable sense, combines the Eulerian and the Lagrangian viewpoint. This is joint work with S. Shkoller.
February
10	Canceled
February
17	Xueke Pu, Brown University KdV limit of Euler-Poisson system
February
24	Yoshiaki Teramoto, Setsunan University Navier-Stokes flow down a vertical flat wall We consider the motion of a viscous incompressible fluid flow down a vertical flat plane under the effect of gravity. We formulate the problem for downward periodic disturbances from the laminar steady flow as an evolution equation in a function space. Under certain assumptions, we show global-in-time existence of solutions of the initial value problem for small disturbances. It is crucial that the operator arising in the linearized problem has compact resolvent operators and generates an analytic semigroup in some function space.
March
2	Yuxi Zheng, Yeshiva University TBA
March
9	Matania Ben-Artzi, Hebrew University TBA
March
16	Miles Wheeler, Brown University
March
23
March
30	Spring break
April
6	Vlad Vicol, University of Chicago
April
13	Xiaodong Yan, University of Connecticut
April
20
April
27	Yanyan Li, Rutgers University TBA
May
4