Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions

TL;DR

This paper explores the deep connections between fluid dynamics, elasticity, and differential geometry, demonstrating how geometric methods can address nonlinear PDEs in continuum mechanics, especially in two dimensions.

Contribution

It develops a geometric framework linking fluids and elasticity with isometric embeddings, applying Gauss-Codazzi equations to analyze nonlinear PDEs in continuum mechanics.

Findings

Geometric theory maps fluid and elastic equations into a differential geometric setting.

Examples demonstrate the application of isometric embedding theory to Euler and elasticity equations.

Results suggest geometric methods can inform admissibility criteria for nonlinear conservation laws.

Abstract

We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss-Codazzi equations through examples for the Euler equations for fluids and the Euler-Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.