Avron Douglis (19181995) received an AB degree in economics from the University of Chicago in 1938. After working as an economist for three years and serving in World War II he began graduate studies in mathematics at New York University. He received his doctorate in 1949 under the direction of Richard Courant. He held a oneyear postdoctoral appointment at the California Institute of Technology, and then returned to New York University as an assistant and then associate professor. In 1956 he accepted an appointment as associate professor at the University of Maryland, where he remained for the rest of his career, except for visiting appointments at the Universities of Minnesota, Oxford, and Newcastle upon Tyne. He was promoted to full professor in 1958 and became an emeritus in 1988.
Avron Douglis's research, noted for its depth, precision, and richness, covered the entire range of the theory of partial differential equations: linear and nonlinear; elliptic, parabolic, and hyperbolic. The famous papers he had written with S. Agmon and L. Nirenberg are among the most frequently cited in all of mathematics.
The Avron Douglis Library is housed in the department.
The Avron Douglis Lectures were established by the family and friends of Avron Douglis to honor his memory. Each academic year it brings to Maryland a distinguished expert to speak on a subject related to partial differential equations.
The lectures are held at 3:00 p.m. in room 3206 in the Department of Mathematics, unless noted otherwise below.
 April 19, 2013

TopologyPreserving Diffusion of DivergenceFree Vector Fields
Yann Brenier
École PolytechniqueThe usual heat equation is not suitable to preserve the topology of divergencefree vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, one can find examples of topologypreserving diffusion equations for divergencefree vector fields. They are very degenerate since they admit all stationary solutions to the Euler equations of incompressible fluids as equilibrium points. For them, we provide a suitable concept of ”dissipative solutions”, which shares common features with both P.L. Lions’ dissipative solutions to the Euler equations and the concept of ”curves of maximal slopes”, à la De Giorgi, recently used by Gigli and collaborators to study the scalar heat equation in very general metric spaces. We show that the initial value problem admits global "dissipative" solutions (at least for two space dimensions) and that they are unique whenever they are smooth.
 February 8, 2012

On the rigidity of black holes
Sergiu Klainerman
Princeton UniversityThe rigidity conjecture states that all regular, stationary solutions of the Einstein field equations in vacuum are isometric to the Kerr solution. The simple motivation behind this conjecture is that one expects, due to gravitational radiation, that general, dynamic, solutions of the Einstein field equation settle down, asymptotically, into a stationary regime. A well known result of Carter, Robinson and Hawking has settled the conjecture in the class of real analytic spacetimes. The assumption of real analyticity is however very problematic; there is simply no physical or mathematical justification for it. During the last five years I have developed, in collaboration with A. Ionescu and S. Alaxakis, a strategy to dispense of it. In my lecture I will these results and concentrate on some recent results obtained in collaboration with A. Ionescu.
 February 25, 2011

Mathematical Strategies for Real Time Filtering of Turbulent Dynamical Systems
Andrew Majda
Courant Institute of Mathematical Sciences  New York UniversityAn important emerging scientific issue in many practical problems ranging from climate and weather prediction to biological science involves the real time filtering and prediction through partial observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom as well as the statistical accuracy of various strategies to cope with the .curse of dimensions.. The speaker and his collaborators, Harlim (North Carolina State University), Gershgorin (CIMS Post doc), and Grote (University of Basel) have developed a systematic applied mathematics perspective on all of these issues. One part of these ideas blends classical stability analysis for PDE's and their finite difference approximations, suitable versions of Kalman filtering, and stochastic models from turbulence theory to deal with the large model errors in realistic systems. Many new mathematical phenomena occur. Another aspect involves the development of test suites of statistically exactly solvable models and new NEKF algorithms for filtering and prediction for slowfast system, moist convection, and turbulent tracers. Here a stringent suite of test models for filtering and stochastic parameter estimation is developed based on NEKF algorithms in order to systematically correct both multiplicative and additive bias in an imperfect model. As briefly described in the talk, there are both significantly increased filtering and predictive skill through the NEKF stochastic parameter estimation algorithms provided that these are guided by mathematical theory. The recent paper by Majda et al (Discrete and Cont. Dyn. Systems, 2010, Vol. 2, 441486) as well as a forthcoming introductory graduate text by Majda and Harlim (Cambridge U. Press) provide an overview of this research.
 April 24, 2009

The global behavior of solutions to critical nonlinear dispersive and wave equations
Carlos E. Kenig
University of ChicagoIn this lecture we will describe a method (which I call the concentrationcompactness/rigidity theorem method) which Frank Merle and I have developed to study global wellposedness and scattering for critical nonlinear dispersive and wave equations. Such problems are natural extensions of nonlinear elliptic problems which were studied earlier, for instance in the context of the Yamabe problem and of harmonic maps. We will illustrate the method with some concrete examples and also mention other applications of these ideas.
 April 25, 2008

Surface Waves and Images
Joseph B. Keller
Stanford University  March 30, 2007

Steady Water Waves: Theory and Computation
Walter Strauss
Brown University  September 30, 2005

A New Perspective on Motion by Curvature
Robert V. Kohn
Courant Institute of Mathematical Sciences, New York University  April 15, 2005

Conservation Laws and Some Consequences
Cathleen Synge Morawetz
Courant Institute of Mathematical Sciences, New York University  March 5, 2004

Hyperbolic Conservation Laws with Dissipation
Constantine Dafermos
Brown University, Division of Applied Mathematics  October 8, 2002

Topology and Sobolev Spaces
Haim Brezis
Universite de Paris VI, Insitiut Universitaire de France, and Rutgers University  April 12, 2002

NavierStokes and Other Supercritical Equations
Vladmir Sverak
University of Minnesota  April 20, 2001

Shock Wave Theory
TaiPing Liu
Academia Sinica, Taiwan & Stanford University  March 31, 2000

Effective Hamiltonians
Lawrence C. Evans
University of California, Berkeley  April 23, 1999

Some remarks on homogenization
Luis Caffarelli
University of Texas, Austin  April 17, 1998

An Example of DiffusionInduced Blowup of a Parabolic System
Hans Weinberger
University of Minnesota  April 4, 1997

The Zero Dispersion Limit
Peter Lax
Courant Institute  May 9, 1996

Degree Theory Beyond Continuous Maps
Louis Nirenberg
Courant Institute
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