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 Fall 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.

Fall 2012

September
14	Luen-Chau Li, PSU On the long time asymptotics of the Camassa-Holm (CH) equation The Camassa-Holm equation is an integrable, nonlinear PDE which models shallow water waves. In this talk, we will discuss several results on the long time asymptotics of the dispersionless case, for which the Riemann-Hilbert approach does not apply.
September
21	Gunduz Caginalp, University of Pittsburgh Generalized Phase Field Models and Interface Problems Problems arising from the study of an interface between two phases such as liquid and solid have been of great interest to mathematicians for centuries. Historically, these problems were modeled using an interface between the two phases (often called the Classical Stefan Model). The phase field approach involves an interfacial region with finite thickness and a smooth transition function. This talk will review the development of this approach and the relationship between the phase field and the free boundary (or sharp interface approaches). Recent developments in collaboration with X. Chen, Ch. Eck and E. Esenturk have extended these results to include non-local interactions with anisotropy. Using differential geometry, combinatorics and asymptotic analysis, we obtain several results: (i) a phase field model that converges more rapidly (as a function of interface thickness) to the sharp interface limiting model, (ii) a mathematically elegant expression for the interface relation when there are anisotropic interactions, (iii) incorporation of non-local microscopic interactions.
September
28	Andrew Comech, Texas A&M Linear instability of solitary waves in nonlinear Dirac equation We study the linear instability of solitary wave solutions to the nonlinear Dirac equation (known to physicists as the Soler model). That is, we linearize the equation at a solitary wave and examine the presence of eigenvalues with positive real part. We show that the linear instability of the small amplitude solitary waves is described by the Vakhitov-Kolokolov stability criterion which was obtained in the context of the nonlinear Schroedinger equation: small solitary waves are linearly unstable in dimensions 3, and generically linearly stable in 1D. A particular question is on the possibility of bifurcations of eigenvalues from the continuous spectrum; we address it using the limiting absorption principle and the Hardy-type estimates. The method is applicable to other systems, such as the Dirac-Maxwell system. Some of the results are obtained in collaboration with Nabile Boussaid, Universite de Franche-Comte, and Stephen Gustafson, University of British Columbia.
October
5	Cancelled
October
12	Luis Silvestre, U of Chicago On the continuity of solutions to drift-diffusion equations We study parabolic equations which consist on a drift term (a vector field times the gradient) plus a diffusion term (either the laplacian or the fractional Laplacian). We analyze what assumptions on the drift would assure that the solution remains continuous for positive time. A particularly interesting case is when the drift is a divergence free vector field, since these appear frequently in equations from fluid mechanics. Assuming that the drift is bounded respect to some norm which is invariant by the scaling of the equation gives Holder continuity estimates in many cases. We will prove that when this scaling condition is violated, discontinuities can form in finite time, even if the drift is divergence free. A notable exception is for an equation with classical diffusion (with the usual Laplacian), in 2 space dimensions, and a drift which is independent of time. For that case a modulus of continuity is obtained for any divergence free drift in $L^1$ (which is highly supercritical).
October
19	Alessio Figalli, University of Texas at Austin and MIT On the regularity of optimal transport maps Knowing whether optimal maps are smooth or not is an important step towards a qualitative understanding of them. In the 90's Caffarelli developed a regularity theory on R^n for the quadratic cost, which was then extended by Ma-Trudinger-Wang and Loeper to general cost functions which satisfy a suitable structural condition. Unfortunately, this condition is very restrictive, and when considered on Riemannian manifolds with the cost given by the squared distance, it is satisfied only in very particular cases. Hence the need to develop a partial regularity theory: is it true that optimal maps are always smooth outside a "small" singular set? The aim of this talk is to first review the "classical" regularity theory for optimal maps, and then describe some recent results about their partial regularity.
October
26	Michael Loss, Georgia Tech The Kac Master Equation; a review In 1956 Mark Kac published his paper ``Foundations of kinetic theory''. In this work he laid out a program for studying systems of randomly colliding particles. He introduced the notion of propagation of chaos and stated his conjecture concerning the gap. I'll review results about this model in particular about the gap, approach to equilibrium in entropy and new results about models that have velocity dependent scattering cross sections. I will also mention some recent work on a Kac-type master equation that describes colliding particles that, in addition, interact with a thermal bath.
November
2	Hermano Frid, IMPA TBA
November
9 (Colloquium at 12pm, Room 110, 182 George)	Nicolai Krylov, UMN Some new results in the theory of fully nonlinear, second order elliptic equations I will be talking about fully nonlinear equations with discontinuous coefficients and about a progress in studying them which occurred quite recently since 2010.
November
9	Roman Shvydkoy, UIC Energetics of the Euler equation and self-similar blow-up The existence of self-similar blow-up for the viscous incompressible fluids was a classical question settled in the seminal of works of Necas, et al and Tsai in the 90'. The corresponding scenario for the inviscid Euler equations has not received as much attention, yet it appears in many numerical simulations, for example those based on vortex filament models of Kida's high symmetry flows. The case of a homogeneous self-similar profile is especially interesting due to its relevance to other theoretical questions such the Onsager conjecture or existence of Landau type solutions. In this talk we give an account of recent studies demonstrating that a self-similar blow-up is unsustainable the Euler system under various weak decay assumptions on the profile. We will also talk about general energetics of the Euler system that, in part, is responsible for such exclusion results.
November
16	Dehua Wang, U of Pittsburgh TBA
November
30	Cancelled
December
7	Charles Smart, MIT Regularity of fully nonlinear elliptic equations without uniform ellipticity I will discuss joint work with Scott Armstrong on the regularity of fully nonlinear elliptic equations when the usual uniform upper bound on the ellipticity is replaced by a bound on its $L^d$ norm, where $d$ is the dimension of the ambient space. Our estimates refine the classical theory and require several new ideas that we believe are of independent interest. As an application, we prove homogenization for a class of stationary ergodic strictly elliptic equations.