Dynamical elastic bodies in Newtonian gravity

TL;DR

This paper proves the well-posedness of the initial value problem for a self-gravitating elastic body with free boundary in Newtonian gravity, establishing conditions for regular solutions.

Contribution

It demonstrates the existence of solutions for elastic bodies under gravity with free boundaries, using a fully nonlinear elliptic-hyperbolic system framework.

Findings

Well-posedness of the initial value problem established

Existence of solutions near relaxed configurations for small Newton's constant

Compatibility conditions for initial data are characterized

Abstract

Well-posedness for the initial value problem for a self-gravitating elastic body with free boundary in Newtonian gravity is proved. In the material frame, the Euler-Lagrange equation becomes, assuming suitable constitutive properties for the elastic material, a fully non-linear elliptic-hyperbolic system with boundary conditions of Neumann type. For systems of this type, the initial data must satisfy compatibility conditions in order to achieve regular solutions. Given a relaxed reference configuration and a sufficiently small Newton's constant, a neigborhood of initial data satisfying the compatibility conditions is constructed.