The Maxey-Riley Equation: Existence, Uniqueness and Regularity of Solutions

The Maxey--Riley equation describes the motion of an inertial (i.e., finite-size) spherical particle in an ambient fluid flow. The equation is a second-order, implicit integro-differential equation with a singular kernel, and with a forcing term that blows up at the initial time. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving local existence and uniqueness of weak solutions. For certain initial velocities between the particle and the fluid, the results extend to strong solutions. We also prove continuous differentiability of the weak and strong solutions with respect to their initial conditions. This justifies the search for coherent structures in inertial flows using the Cauchy--Green strain tensor.