Virial Theorems and Equipartition of Energy for Water Waves

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 5285-5329, August 2024.
Abstract. We study several different aspects of the energy equipartition principle for water waves. We prove a virial identity that implies that the potential energy is equal, on average, to a modified version of the kinetic energy. This is an exact identity for the complete nonlinear water-wave problem, which is valid for arbitrary solutions. As an application, we obtain nonperturbative results about the free-surface Rayleigh–Taylor instability, for any nonzero initial data. We also derive exact virial identities involving higher order energies. We illustrate these results by an explicit computation for standing waves. As an aside, we prove trace inequalities for harmonic functions in Lipschitz domains which are optimal with respect to the dependence in the Lipschitz norm of the graph.