Mathematical and numerical analysis of the rayleigh-plesset and the keller equations

In the present paper, we conduct mathematical analysis on the Rayleigh- Plesset and the Keller equations, ordinary differential equations of the second order widely used for describing motions of a spherically symmetric single bubble. We show that these equations admit structures of the Hamiltonian system with respect to a physically reasonable energy function perturbed by dissipation and obtain the asymptotic behavior of the solutions. Making use of this structure, we rewrite the equations into gradient systems and develop numerical codes which properly inherit conservation or dissipation of the energy from the original differential equations following the discrete gradient method.