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

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.