# Existence theory for magma equations in dimension two and higher

## TL;DR

This paper proves local well-posedness and the existence of solitary wave solutions for a nonlinear magma equation in two or more dimensions, advancing the mathematical understanding of melt dynamics in geophysics.

## Contribution

It establishes the first rigorous mathematical results on well-posedness and solitary waves for the magma equation in higher dimensions, extending previous numerical observations.

## Key findings

- Proved local in time well-posedness in 2D and higher
- Established existence of solitary wave solutions in multiple dimensions
- Extended mathematical theory to complex geophysical models

## Abstract

We examine a degenerate, dispersive, nonlinear wave equation related to the evolution of partially molten rock in dimensions two and higher. This simplified model, for a scalar field capturing the melt fraction by volume, has been studied by direct numerical simulation where it has been observed to develop stable solitary waves. In this work, we prove local in time well-posedness results for the time dependent equation, on both the whole space and the torus, for dimensions two and higher. We also prove the existence of the solitary wave solutions in dimensions two and higher.