Global exponential stability for the three-dimensional Navier-Stokes equations on hyperbolic space

TL;DR

This paper proves the global exponential stability of solutions to the 3D Navier-Stokes equations on hyperbolic space, highlighting the role of negative curvature in spectral gap and decay properties.

Contribution

It establishes the existence of unique global solutions with exponential decay on hyperbolic space, contrasting with algebraic decay in flat space, and analyzes the influence of curvature on stability.

Findings

Solutions decay exponentially with rate μλ_D^{(3)} on hyperbolic space.

Spectral gap due to negative curvature enables exponential decay.

Supercritical L^2 norm obstruction linked to local heat kernel scaling.

Abstract

We prove that the three-dimensional incompressible Navier-Stokes equations with the deformation Laplacian on hyperbolic 3-space $\HH^3$ admit a unique global mild solution for sufficiently small initial data in $L^3(\HH^3)$, and that this solution decays exponentially to zero. The exponential decay rate is $\mu\lambda_\Def^{(3)}$, where $\mu$ is the dynamic viscosity and $\lambda_\Def^{(3)} = 26/9$ is the effective spectral gap of the deformation Laplacian in $L^3$. On flat $\R^3$, the corresponding Kato-type result gives only algebraic decay. The exponential stability is a macroscopic consequence of the spectral gap provided by negative curvature. We also show that the $L^2$ norm is supercritical on $\HH^3$ (as on $\R^3$), with the obstruction arising from the local ultraviolet scaling of the heat kernel, which is insensitive to global geometry. The boundary between what curvature can and cannot improve is located exactly: the Fujita-Kato integral has a scaling exponent $1/2 - 3/(2p)$ that depends only on the integrability of the initial data, not on the geometry of the manifold. For $p \geq 3$ (the Kato critical space), the integral is bounded and the spectral gap contributes exponential time decay. For $p < 3$, the integral diverges at $t = 0$ (and strictly diverges for all $t>0$ when $p \le 2$) regardless of the curvature.