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 2013, 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.

Please also visit the Seminars webpage on the Applied Mathematics website.

Spring 2013

February
1	Xuwen Chen (Brown) On the Rigorous Derivation of the 2D Cubic Nonlinear Schroedinger Equation from 3D Quantum Many-Body Dynamics In this talk, I will talk about my recent joint work with J. Holmer on how the 2D cubic NLS arises from the 3D quantum N-body dynamics describing a dilute bose gas with strong confining in one direction. This could be viewed as the quantum version of the Kac's program. But this 3D to 2D phenomenon is a quantum behavior and does not exist in the classical (Boltzmann) setting. In physics, this corresponds to the justification of the Gross-Pitaevskii limit in Bose-Einstein condensation. The main difficulty is that the corresponding BBGKY hierarchy contains a diverging coefficient as the strength of the confining potential tends to infinity.
February
6 (Special time: 4:00-5:00pm at B&H Room 190)	Joel Spruck (Johns Hopkins University) The half-space property and entire positive minimal graphs in MxR An important question is to understand when two natural objects, for example two complete minimal hypersurfaces S1, S2 in a Riemannian manifold N, must intersect. In this talk we consider this question when N = M x R where M is a complete n dimensional Riemannian manifold, S1 = M x {0} and S2 is a properly what conditions on M imply that S = S2 is the totally geodesic slice M x {c} for some positive c. The celebrated theorem of Bombieri, De Giorgi and Miranda, which says that an entire positive minimal graph over Rn must be a totally geodesic slice is perhaps the first such result. Another foundational result is the Hoffman-Meeks half-space theorem which states that if S is a properly immersed minimal surface in R3=R2 x R+, then S = R2 x {c}, c>=0. Since there are rotationally invariant minimal hypersurfaces (catenoids) in Rn+1, n > 2, that are bounded above and below, the Hoffman-Meeks theorem is false for M = Rn, n>2.
February
7 (Special time: 12-1pm at B&H 190)	Luis Caffarelli, (University of Texas-Austin) The homogenization of Surfaces and Edges We will discuss, in several different examples, a surface minimizing area in a highly oscillatory medium, the edge of a drop siting on a rough plane, a moving front, the existence of homogenization limits, that is: if when seen from afar, they become smooth surfaces satisfying an "effective" equation".
February
8 (special time: 11-12pm, DAM 110)	Irene Gamba (University of Texas-Austin) TBA
March
1	Zaher Hani (NYU) Scattering for nonlinear dispersive equations in the presence of a potential. Questions related to the asymptotic behavior of nonlinear dispersive equations in the presence of a potential term are of great interest both for mathematical and physical reasons. Our main concern will be equations with low-degree nonlinearities, namely below the Strauss exponent threshold, for which classical energy and decay methods fail to suffice. For this, we we use the spectral theory of the operator $H=-\Delta+V$ to develop a space-time resonance analysis adapted to the inhomogeneous setting. A key ingredient in this setup is the development of a sufficiently comprehensive multilinear harmonic analysis in the context of the corresponding distorted Fourier transform. This turns out to exhibit several intriguing differences in comparison to the unperturbed Euclidean setting (no matter how small V is). As a first application, we treat the case of a quadratic nonlinear Schrodinger equation on $\R^3$. This is joint work with Pierre Germain and Samuel Walsh (Courant Institute, NYU).
March
8	Naoyuki Ichihara (Hiroshima University) Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control We are concerned with nonlinear additive eigenvalue problems for viscous Hamilton-Jacobi equations which appear in stochastic ergodic control. Certain qualitative properties of principal eigenvalues and associated eigenfunctions are studied. Such analysis plays a key role in studying the recurrence and transience of feedback diffusions for the corresponding stochastic control problems. Our results can be regarded as a nonlinear extension of the criticality theory for Schrodinger operators with decaying potentials.
March
22	Tianling Jin (U of Chicago) Solutions of some Monge-Amp\`ere equations with singularities We will give a new proof of a celebrated theorem of J\"orgens which states that every classical convex solution of det(Hess u)=1 in R^2 has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions some degenerate Monge-Amp\`ere equations. We will also discuss some results on existence, regularity, classification, and asymptotical behaviors of solutions of some Monge-Amp\`ere equations with isolated and line singularities. These are joint works with Jingang Xiong.
March
29	Juhi Jang (UC Riverside) Stability theory of polytropic gaseous stars I will discuss stability theory of Lane-Emden equilibrium stars under Euler-Poisson or Navier-Stokes-Poisson system. A linear stability can be characterized by the adiabatic exponent. A nonlinear instability will be also discussed.
April
19	Yifeng Yu (UCI) Flame front quenching and homogenization of a non-convex and non-coercive Hamilton Jacobi equation I will talk about periodic homogenization of a highly non-coercive and non-convex Hamilton-Jacobi equation from the modeling of turbulent combustion in 2d cellular flow(G-equation with a flow strain rate term). The main goal is to understand the effect of strain rate as flow intensity increases. Our main result says that when the flow intensity is sufficiently high, the flame front ceases to propagate forward. This is a joint work with Jack Xin.
May
3	Benedetto Piccoli (Rutgers University) Conservation Laws on Networks and Collective Dynamics
May
10	TBA

PDE Seminar Calendar

Spring 2013