Research

"The laws of nature should be expressed in beautiful equations."
- Paul Dirac (1902-1984)

My areas of interest and study include mathematical biology, nonlinear partial differential equations (PDEs), and incompressible fluid dynamics.

The quote by Paul Dirac, a famous mathematician, suggests that the laws of nature are written in the language of mathematics. Algebra is sufficient to solve many static problems, but the most interesting natural phenomena involve change and are described only by equations that relate changing quantities. These equations are called differential equations. The study of differential equations uses calculus, linear algebra, the study of functions, analysis, numerical methods and other areas of mathematics to examine problems rooted in physics, the biological sciences, and engineering. There are many exciting undergraduate projects (see the section below) combining differential equations and disciplines outside of mathematics. If you are interested in more information, please contact me!


Publications
1. Peirce, J. Local well-posedness of the anisotropic Lagrangian averaged Navier-Stokes equations with Stokes projector viscosity. Comm. Partial Differential Equations, 31 (2006), no. 8, 1139--1149.
2. Peirce, J., Well-posedness of the three-dimensional Lagrangian averaged Navier-Stokes equations. 2004. (Ph.D. Dissertation)
3. Coutand, D., Peirce, J., Shkoller, S. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Commun. Pure Appl. Anal. 1, (2002), no.1, 35--50.

Currrent Research

Undergraduate Research Advising:

Fall-Spring 2007-2008. I advised math major Trever Hallock on an undergraduate research project on the motion of a vortex line in an averaged velocity field. One approach taken to understanding turbulence is the development of mathematical models for the movement of vortex filaments. A vortex filament is a thin tube created when the fluid rotates or spins very intensely about a single curve. When a filament rotates, it changes the fluid movement around it. The induced motion of the fluid then transports the filament. The resulting equation of motion for a vortex filament depends on the fluid velocity it has created. A good example of a vortex filament is a tornado. The rotation of the tornado about its axis causes the air around it to move quickly. The air motion causes high winds that change the shape of the tornado and puts the tornado into motion along a path. Trever derived an equation describing the motion of a vortex line in a fluid domain with an average velocity field. He received a research stipend and presented his work at the 2008 UW-L Celebration of Research and Creativity Day. Trever is currently a student at University of California, Berkeley.

Summer 2005. I advised Devin Bickner on an undergraduate research project. Devin used partial differential equations to model the motion of a plucked guitar string. He proposed an equation that included the effects of dampening and proved that the total energy of this new model decreases as we expect from a real guitar string. Devin solved the general partial differential equation, applied it to the plucked string model, and used Mathematica to "pluck the string" and hear the solution. Devin presented his work at the Pi Mu Epsilon Conference at St. Norbert College in Fall 2005. Devin has decided to continue his study of mathematics in a Ph.D. program at Iowa State University.

Resources