Elliptic Regularization and the Solvability of Self-Propelled Locomotion Problems

Abstract

Humans have been fascinated by the wonders of animal locomotion since antiquity. The effortless grace of birds in flight, and the silent glide of fish through water exemplify the elegance possible with control over the local fluid flow. While we have been free to observe the poetry in these motions for centuries, the ability to predict and mimic these motions with any degree of accuracy is a much more recent development. Advances in computing hardware and our numerical simulation capabilities have opened the door to the analysis and prediction of the motion of complex shapes in fluid environments. As we continue to develop more depth and sophistication in our ability to simulate the mathematical models of fluid mechanics, we must be sure to maintain an awareness of the existence and smoothness of the exact solutions to our mathematical models. In this paper we present a very brief overview of some techniques from partial differential equations which can be used to analyze physically relevant problems in animal locomotion. The ideas used here give important insight into situations where control systems for ordinary differential equations may be used to control the trajectories of vehicles through fluid environments.