A weak solution for a point mass camphor motion

Abstract

The model system of equations which describes the self-propelled motion of point mass objects driven by camphor is a diffusion equation coupled with a system of nonlinear ordinary differential equations. If the objects have masses, then the motion of objects becomes very complicated when some of the objects hit the boundary of a water surface or collide each other. To avoid such complexity and try to get some general perspective for the motion, it is convenient to consider point mass objects, i.e., objects without areas. We give an existence of a weak solution of this model system by giving an a priori estimate for the solution. The key to this estimate is the choice of a special test function. This is a first step toward analyzing collective motion of point mass camphors. As far as we know, this result is the first result on the existence of a weak solution for this system.