From the Nash--Kuiper Theorem of Isometric Embeddings to the Euler Equations for Steady Fluid Motions: Analogues, Examples, and Extensions

TL;DR

This paper explores the deep connections between isometric embeddings of surfaces and steady fluid dynamics, establishing mappings from geometric PDEs to fluid equations, and analyzing special solutions and potential multi-dimensional extensions.

Contribution

It introduces a novel mapping linking isometric embedding equations to steady Euler equations, providing new insights into geometric-fluid dynamics analogies and solutions.

Findings

Constructed a mapping from second fundamental form to fluid variables.

Analyzed special solutions like Chaplygin gas on tori.

Discussed potential extensions to higher dimensions.

Abstract

Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface $(M,g)$ isometrically embedded in $\mathbb{R}^3$, we construct a mapping which sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on $(M,g)$. From the PDE perspectives, this mapping sends solutions to the Gauss--Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori, as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multi-dimensions.