Nonuniqueness of solutions of the Navier-Stokes equations on negatively curved Riemannian manifolds

TL;DR

This paper demonstrates that on certain negatively curved Riemannian manifolds, solutions to the Navier-Stokes equations are not unique, challenging assumptions about solution behavior in geometric analysis.

Contribution

It shows nonuniqueness of Navier-Stokes solutions on specific negatively curved manifolds, extending previous geometric analysis results.

Findings

Existence of non-unique solutions on negatively curved manifolds

Construction of manifolds with specific harmonic form properties

Implications for Navier-Stokes solution theory in geometric contexts

Abstract

In a well-known work, M. Anderson constructed a Hadamard manifold $(M^n, g)$ which carries non-zero $L^2$ harmonic $p$-forms when $p \neq n/2$, thus disproving the Dodziuk-Singer conjecture. In this paper, we use the manifold $(M^3, g)$ in order to solve another problem in geometric analysis, namely the nonuniqueness of solutions of Leray-Hopf type of the Navier-Stokes equations.