On a Hamiltonian PDE arising in Magma Dynamics

TL;DR

This paper introduces a new Hamiltonian PDE model for magma dynamics in Earth's interior, proving the stability of solitary waves and demonstrating the existence of compactons, thereby extending understanding of wave behavior in geophysical flows.

Contribution

It presents a novel Hamiltonian PDE derived from magma equations, proves the nonlinear stability of solitary waves, and shows the existence of compactons in this context.

Findings

Solitary waves are nonlinearly stable as constrained local minimizers.

The PDE admits compacton solutions with finite support.

Global well-posedness extends to large data near solitary waves.

Abstract

In this article we discuss a new Hamiltonian PDE arising from a class of equations appearing in the study of magma, partially molten rock, in the Earth's interior. Under physically justifiable simplifications, a scalar, nonlinear, degenerate, dispersive wave equation may be derived to describe the evolution of $\phi$, the fraction of molten rock by volume, in the Earth. These equations have two power nonlinearities which specify the constitutive realitions for bulk viscosity and permeability in terms of $\phi$. Previously, they have been shown to admit solitary wave solutions. For a particular relation between exponents, we observe the equation to be Hamiltonian; it can be viewed as a generalization of the Benjamin-Bona-Mahoney equation. We prove that the solitary waves are nonlinearly stable, by showing that they are constrained local minimizers of an appropriate time-invariant Lyapunov functional. A consequence is an extension of the regime of global in time well-posedness for this class of equations to (large) data, which include a neighborhood of a solitary wave. Finally, we observe that these equations have {\it compactons}, solitary traveling waves with compact spatial support at each time.