Virial theorems and equipartition of energy for water-waves

TL;DR

This paper establishes exact virial identities for water waves, demonstrating energy equipartition and applying these results to phenomena like bubble formation in Rayleigh-Taylor instability, with broader implications for nonlinear equations.

Contribution

It introduces exact virial identities for nonlinear water wave equations, providing new insights into energy distribution and stability analysis.

Findings

Potential energy equals modified kinetic energy on average for water waves

Non-perturbative results justify bubble formation in Rayleigh-Taylor instability

Derived optimal trace inequalities for harmonic functions in Lipschitz domains

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 non-perturbative results justifying the formation of bubbles for the free-surface Rayleigh-Taylor instability, for any non-zero initial data. We also derive exact virial identities involving higher order energies. The fact that such exact identities are valid for nonlinear equations is new and general: as explained in a companion paper, similar identities can be derived for many other nonlinear equations. We illustrate this result by an explicit computation for standing waves. As side results, 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.