Differential and Integral Equations

A weak solution for a point mass camphor motion

Jishan Fan, Masaharu Nagayama, Gen Nakamura, and Mamoru Okamoto

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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.

Article information

Differential Integral Equations, Volume 33, Number 7/8 (2020), 431-443.

First available in Project Euclid: 14 July 2020

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Primary: 35K99: None of the above, but in this section 35Q70: PDEs in connection with mechanics of particles and systems


Fan, Jishan; Nagayama, Masaharu; Nakamura, Gen; Okamoto, Mamoru. A weak solution for a point mass camphor motion. Differential Integral Equations 33 (2020), no. 7/8, 431--443. https://projecteuclid.org/euclid.die/1594692056

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