Well-posedness for the Navier-Stokes equations in Morrey spaces on non-compact manifolds

TL;DR

This paper establishes well-posedness of the Navier-Stokes equations on non-compact manifolds with negative curvature using Morrey spaces, providing new dispersive estimates and accommodating rough initial data.

Contribution

It introduces Morrey space analysis for Navier-Stokes on curved manifolds, extending well-posedness results to broader initial data classes and non-compact geometries.

Findings

Dispersive and smoothing estimates for heat semigroups in Morrey spaces

Local and global well-posedness results under curvature and smallness conditions

Extension to Ricci-flat manifolds and new classes of initial data

Abstract

We analyze the incompressible Navier-Stokes equations on a class of non-compact Riemannian manifolds within the framework of Morrey spaces. Assuming bounded geometry together with negative Ricci and sectional curvature (e.g., hyperbolic spaces), we establish dispersive and smoothing estimates for the heat semigroups associated with the Beltrami, Bochner and Hodge Laplacians in Morrey spaces, as well as for the Riesz transform. In particular, the presence of negative curvature yields improved large-time decay compared to the Euclidean setting. These estimates are of independent interest and enable us to construct solutions in time-weighted spaces of Kato type, leading to local-in-time well-posedness on a broad class of non-compact manifolds and global one in the case of Einstein manifolds. In the latter setting, we assume a smallness condition on the initial data in Morrey norms, which are weaker than $L^{p}$-norms and thus allow for certain classes of large $L^{p}$-data. We also discuss extensions to Ricci-flat manifolds. Our results introduce a new class of non-decaying and rough initial data for the Navier-Stokes equations on manifolds, extending previous works in Lebesgue and Sobolev spaces.