# Research: Analysis and Differential Equations

The analysis/differential equations group conducts research in theory and applications of ordinary and partial differential equations and dynamical systems. The areas of research include: differential equations (ODEs and PDEs), difference equations, dynamical systems, ergodic theory, fluid dynamics, long time behavior of dynamical systems, modeling in mathematical biology, nonlinear PDEs and applications,stochastic ODEs and PDEs, fluid dynamics (Navier-Stokes, Euler, and Boussinesq equations).

###### Tenured and tenure-track faculty:

- Friedlander, Susan (fluid mechanics, partial differential equations)
- Haydn, Nicolai (dynamical systems, ergodic theory)
- Jang, Juhi (analysis and PDEs, compressible NSE and Euler equations, kinetic theory, plasma physics, astrophysics)
- Kukavica, Igor (elliptic and parabolic partial differential equations, nonlinear dynamics, fluid dynamics)
- Newton, Paul (nonlinear dynamics, fluid dynamics, vortex dynamics, game theory, cancer metastasis research, Markov chains)
- Sacker, Robert (ordinary differential equations, difference equations, infinite dimensional dynamical systems, applied mathematics, mathematical biology)
- Ziane, Mohammed (partial differential equations, geophysical dynamics, Navier-Stokes equations, stochastic partial differential equations)

###### Other faculty with interests in analysis/differential equations:

###### Postdoctoral associates:

- Ryals, Brian (dynamical systems)

###### Activities:

- CAMS Colloquium
- Dynamical Systems Seminar

- Math 725, Graduate Analysis Seminar

###### Graduate students:

- Camliyur, Guher (partial differential equations, fluid dynamics); advisor: Kukavica
- Ugurlu, Kerem (stochastic partial differential equations, fluid dynamics); advisor: Ziane
- Wang, Fei (partial differential equations, harmonic analysis, fluid dynamics); advisor: Kukavica
- Yang, Fan (dynamical systems); advisor: Haydn

###### Recent PhD graduates:

- Torres, Juan (Kukavica), Space-time decay of solutions of the Navier-Stokes equations, 2006
- Glatt-Holtz, Nathan (Ziane), Stochastic equations in geophysical fluid dynamics, 2008 (currently: Assistant Professor, Virginia Tech)
- Psiloyenis, Yiannis (Haydn), Mixing conditions and long return times on Markov towers, 2008 (currently: Farmer's insurance)
- Vicol, Vlad (Kukavica), Analyticity and Gevrey class regularity for the Euler equations, 2010 (currently: Assistant Professor, Princeton University)
- Ignatova, Mihaela (Kukavica), Quantitative unique continuation and complexity of solutions to partial differential equations, 2011 (currently: Postdoc, Princeton University)
- Reis, Ednei (Kukavica), Asymptotic expansion for solutions of the Navier-Stokes equations with potential forces, 2011 (currently: Professor Adjunto, Universidade tecnologica federal do Parana, Brasil)
- Ko, Hong Fu Milton (Haydn), Recurrence and Renyi entropy, 2012 (currently: Business Analyst, Sersys salutations Ltd.)
- Ostrovskyi, Vitalii (Newton), Point singularities on 2D surfaces, 2013 (currently: Sr. Data Architect, American Express)
- Williams, Katarzyna (Haydn), Limiting distribution and error terms for the number of visits to balls in mixing dynamical systems, 2013 (currently: teacher, Harvard-Westlake school)
- Pei, Yuan (Kukavica), Certain regularity problems in fluid dynamics, 2014 (currently: Postdoc, University of Nebraska)
- Garcia-Torres, Selenne (Sacker) Structured two-stage population model with migration between multiple patches in a periodic environment, 2013 (currently: lecturer, USC)
- Yang, Fan (Haydn) Entry times statistics on metric spaces, 2015

###### Recent Postdoctoral associates:

- Gupta, Chinmaya (dynamical systems)
- Hu, Weiwei (partial differential equations, control theory)
- Rusin, Walter (PDEs, fluid dynamics)
- Suen, Anthony (partial differential equations, harmonic analysis)
- Tuffaha, Amjad (PDEs, fluid dynamics)

###### Emeritus faculty:

- Bruck, Ronald E. (nonlinear functional analysis, ergodic theory)
- Vorel, Zdenek (functional analysis)