# PDE & Dynamical Systems

Partial differential equations (PDEs) are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory. The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse.

Many of the faculty in the Duke mathematics department conduct research on partial differential equations, using a variety of techniques. Some of this work focusses on the geometric structure of PDEs, on the analytical properties of their solutions, or on the way in which solutions depend on parameters. Some faculty are developing computer algorithms for approximating solutions to complicated PDEs. Other faculty are developing new mathematical models based on PDEs in the context of applications in fluid mechanics, photonics, materials science, and biology.

## Faculty

#### J. Thomas Beale

###### Professor Emeritus of Mathematics

**Keywords in this area**

boundary integral methods, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow

#### Robert Bryant

###### Phillip Griffiths Professor of Mathematics

**Keywords in this area**

differential geometry, holonomy, exterior differential systems, integrability, symplectic geometry

#### Mark Haskins

###### Professor of Mathematics

**Keywords in this area**

Nonlinear systems of elliptic PDEs. Regularity theory of calibrated currents. Analysis on noncompact complete spaces and on incomplete or singular spaces.

#### Gregory Joseph Herschlag

###### Assistant Research Professor of Mathematics

**Keywords in this area**

biological modeling, physical modeling, fluids flow across dynamical channels, kinetic equations, surface catalysis, molecular dynamics, stochastic boundary conditions

#### Alexander A. Kiselev

###### William T. Laprade Distinguished Professor of Mathematics

**Keywords in this area**

Fluid mechanics, Euler equation, surface quasi-geostrophic equation, singularity formation, small scale creation, mixing efficiency

#### Jianfeng Lu

###### Professor of Mathematics

**Keywords in this area**

multiscale modeling and simulations, numerical analysis, calculus of variations, kinetic equations, Schroedinger equations

#### Jonathan Christopher Mattingly

###### James B. Duke Distinguished Professor

**Keywords in this area**

Stochastic and Random PDEs, Stochastic Dynamical Systems, Malliavin Calculus, Fluid Mechanics, Approximating invariant measures

#### Sayan Mukherjee

###### Professor of Statistical Science

**Keywords in this area**

parameter estimation in dynamical systems

#### James H. Nolen

###### Associate Professor of Mathematics

**Keywords in this area**

reactive diffusion equations & applications, homogenization of partial differential equations, random media, asymptotic analysis

#### Arlie O. Petters

###### Benjamin Powell Distinguished Professor of Mathematics

**Keywords in this area**

gravity, light, mathematical finance

#### Mark A. Stern

###### Professor of Mathematics

**Keywords in this area**

geometric analysis, elliptic partial differential equations

#### Stephanos Venakides

###### Professor of Mathematics

#### Thomas P. Witelski

###### Professor in the Department of Mathematics

**Keywords in this area**

fluid dynamics, nonlinear partial differential equations, dynamical systems, perturbation methods