Calculus of Variations and Geometric Measure Theory

G. Dal Maso - A. DeSimone - M. Morandotti

An existence and uniqueness result for the motion of self-propelled micro-swimmers

created by dalmaso on 02 Jul 2010
modified by morandott on 08 Dec 2011


Published Paper

Inserted: 2 jul 2010
Last Updated: 8 dec 2011

Journal: SIAM J. Math. Anal.
Volume: 43
Number: 3
Pages: 1345-1368
Year: 2011


We present an analytical framework to study the motion of micro-swimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of shape determines uniquely the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding, infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton's equations of motion reduce to the vanishing of the viscous drag force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of its coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution.

Keywords: swimming, low Reynolds number flows, dependence of solutions of PDEs on domain and boundary data, Stokes equations