Avron Douglis (1918-1995) 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 one-year post-doctoral 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.

### April 17, 2024 - Long Time Dynamics of Fluids and Plasmas

Peter Constantin

Princeton University

Fluids interacting with charged particles belong to many realms of scientific endeavor, ranging from electrochemistry and electrophysiology, to astrophysics and nuclear fusion engineering. I will discuss first the case of electrostatic interactions, relevant to electrochemistry and elctrophysiology. The models consist of Navier-Stokes equations for the fluid, coupled to equations describing the evolution of charge densities, with negligible magnetic effects. The resulting systems, Nernst-Planck-Navier-Stokes and electroconvection systems, have a rich dynamical behavior under non-equilibrium boundary conditions, but exhibit entropic relaxation under equilibrium boundary conditions. Mathematically, this relaxation is due to the presence of nontrivial dissipative structures. I will present some of the ideas of proofs of regularity and stability, some of the results about the existence of non-equilibrium states, and the problem of electroneutrality. I will then discuss strong magnetic fields and relaxation to magneto-hydrostatic (MHS) equilibria via viscous and non-dissipative mechanisms. This is a mathematical and potentially numerical approach to reach MHS configurations which are of interest to stellarator fusion design. I will present some of the challenges of this area. Finally, if time permits, I will also briefly discuss problems of relativistic plasma kinetics.

#### About the Speaker

Peter Constantin is the John von Neumann Professor of Mathematics and Applied and Computational Mathematics at Princeton University. Peter Constantin received his B.A and M.A. summa cum laude from the University of Bucharest, Faculty of Mathematics and Mechanics. He obtained his Ph.D. from The Hebrew University of Jerusalem under the direction of Shmuel Agmon. Constantin's work is focused on the analysis of PDE and nonlocal models arising in statistical and nonlinear physics. Constantin worked on scattering for Schrodinger operators, on finite dimensional aspects of the dynamics of Navier-Stokes equations, on blow up for models of Euler equations. He introduced active scalars, and, with Jean-Claude Saut, local smoothing for general dispersive PDE. Constantin worked on singularity formation in fluid interfaces, on turbulence shell models, on upper bounds for turbulent transport, on the inviscid limit, on stochastic representation of Navier-Stokes equations, on the Onsager conjecture. He worked on critical nonlocal dissipative equations, on complex fluids, and on ionic diffusion in fluids. Constantin has advised thirteen graduate students in mathematics, and served in the committee of seven graduate students in physics. He mentored twenty-five postdoctoral associates. Constantin served as Chair of the Mathematics Department of the University of Chicago and is the Director of the Program in Applied and Computational Mathematics at Princeton University. Constantin is a Fellow of the Institute of Physics, a SIAM Fellow, and Inaugural Fellow of the American Mathematical Society, a Fellow of the American Academy of Arts and Sciences and a member of the National Academy of Sciences.

### February 14, 2023 - Internal Waves in 2D Aquaria and Homeomorphisms of the Circle

Maciej Zworski

UC Berkeley

The connections between the formation of internal waves in fluids, spectral theory, and homeomorphisms of the circle were investigated by oceanographers in the 90s and resulted in novel experimental observations (Leo Maas et al, 1997). The specific homeomorphism is given by a ``chess billiard" and has been considered by many authors (Fritz John 1941, Vladimir Arnold 1957, Jim Ralston 1973... ). The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!). I will illustrate the results with numerical and experimental examples and explain how classical concepts such as rotation numbers of homeomorphisms (introduced by Henri Poincare) are related to solutions of the Poincare evolution problem (so named by Elie Cartan). The talk is based on joint work with Semyon Dyatlov and Jian Wang. I will also mention recent progress by Zhenhao Li on the case of irrational rotation numbers.

### March 4, 2020 - Flows of Vector Fields: Classical and Modern

**Camillo De Lellis**IAS, Princeton

Consider a (possibly time-dependent) vector field v on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) Theorem states that, if the vector field v is Lipschitz in space, for every initial datum x there is a unique trajectory γ starting at x at time 0 and solving the ODE.(t) = v(t, γ(t). The theorem looses its validity as soon as v is slightly less regular. However, if we bundle all trajectories into a global map allowing x to vary, a celebrated theory put forward by DiPerna and Lions in the 80's show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for almost every initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov’s h-principle.

### December 8, 2017 - The Threshold Theorems for the Hyperbolic Yang-Mills Equations

Daniel Tataru

UC Berkeley

This talk will aim to provide an overview of a recent series of papers, joint with Sung-Jin Oh, devoted to the energy critical 4+1 dimensional hyperbolic Yang-Mills equation. These papers provide a comprehensive analysis of the large data problem, ultimately providing a proof of the Threshold Conjecture for Yang-Mills, and more. We will cover an array of ideas, ranging from gauge theory to hard core pde estimates to geometry and blow-up analysis.

### November 2, 2016 - Effective Models for Ginzburg-Landau Vortices

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.

### March 31, 2016 - The h-principle in Fluid Mechanics

László Székelyhidi Jr.

Institute of Mathematics, University of Leipzig

It is known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence. In joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov’s h-principle. In this lecture I will explain this connection and outline recent progress concerning Onsager’s conjecture.

### April 9, 2015 - From molecular dynamics to kinetic theory and fluid mechanics

**Laure Saint-Raymond**Harvard and MIT on leave from Ecole Normale Superieure

In his sixth problem, Hilbert asked for an axiomatization of gas dynamics, and he suggested to use the Boltzmann equation as an intermediate description between the (microscopic) atomic dynamics and (macroscopic) fluid models. The main difficulty to achieve this program is to prove the asymptotic decorrelation between the local microscopic interactions, referred to as propagation of chaos, on a time scale much larger than the mean free time. This is indeed the key property to observe some relaxation towards local thermodynamic equilibrium. This control of the collision process can be obtained in fluctuation regimes [1, 2]. In [2], we have established a long time convergence result to the linearized Boltzmann equation, and eventually derived the acoustic and incompressible Stokes equations in dimension 2. The proof relies crucially on symmetry arguments, combined with a suitable pruning procedure to discard super exponential collision trees.

Keywords: system of particles, low density, Boltzmann-Grad limit, kinetic equation, fluid models

[1] T. Bodineau, I. Gallagher, L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres, to appear in Invent. Math. (2015).

[2] T. Bodineau, I. Gallagher, L. Saint-Raymond. From hard spheres to the linearized Boltzmann equation: an L2 analysis of the Boltzmann-Grad limit, in preparation.

### April 17, 2014 - Regularity, blow up, and small scale creation in fluids

**Alexander Kiselev**University of Wisconsin at Madison

The Euler equation of fluid mechanics describes a flow of inviscid and incompressible fluid, and has been first written in 1755. The equation is both nonlinear and nonlocal, and its solutions often create small scales easily and tend to be unstable. I will review some of the background, and then discuss a recent sharp result on small scale creation in solutions of the 2D Euler equation. I will also indicate links to the long open question of finite time blow up for solutions of the 3D Euler equation.

### April 19, 2013 - Topology-Preserving Diffusion of Divergence-Free Vector Fields

**Yann Brenier**École Polytechnique

The usual heat equation is not suitable to preserve the topology of divergence-free vector fields, because it destroys their integral line structure. On the contrary, in the fluid mechanics literature, one can find examples of topology-preserving diffusion equations for divergence-free 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 University

The 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 University

An 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 slow-fast 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, 441-486) 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 Chicago

In this lecture we will describe a method (which I call the concentration-compactness/rigidity theorem method) which Frank Merle and I have developed to study global well-posedness and scattering for critical non-linear dispersive and wave equations. Such problems are natural extensions of non-linear 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 - Navier-Stokes and Other Super-critical Equations

**Vladmir Sverak**

University of Minnesota

### April 20, 2001 - Shock Wave Theory

**Tai-Ping 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 Diffusion-Induced 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