[Paper Review] The incompressible Navier-Stokes equations on non-compact manifolds

Vittoria Pierfelice|arXiv (Cornell University)|Jun 6, 2014

Advanced Mathematical Physics Problems51 references17 citations

TL;DR

This paper establishes dispersive and smoothing estimates for the Bochner Laplacian on non-compact Riemannian manifolds with negative Ricci curvature, particularly hyperbolic spaces, and applies them to prove Fujita-Kato-type global well-posedness results for the incompressible Navier-Stokes equations. It further establishes uniqueness of Leray weak solutions in two dimensions on such manifolds via energy methods and Gronwall's inequality.

ABSTRACT

We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We shall also discuss the uniqueness of Leray weak solutions in the two dimensional case.

Motivation & Objective

• To extend the Kato-Fujita approach to the incompressible Navier-Stokes equations beyond Euclidean space to non-compact Riemannian manifolds with negative Ricci curvature.
• To derive dispersive and smoothing estimates for the Bochner Laplacian on such manifolds, particularly hyperbolic spaces.
• To establish global existence and uniqueness of solutions for the Navier-Stokes system under critical or subcritical initial data conditions on these geometric settings.
• To prove the uniqueness of Leray weak solutions in two dimensions on non-compact manifolds via energy equality and Gronwall-type estimates.

Proposed method

• Derives dispersive and smoothing estimates for the Bochner Laplacian on non-compact Riemannian manifolds with negative Ricci curvature, leveraging curvature-induced decay properties.
• Applies these estimates to the linearized Navier-Stokes system to establish well-posedness in the spirit of the Fujita-Kato fixed-point method.
• Uses the Hodge decomposition and Leray projector to eliminate pressure terms and reduce the system to a semilinear parabolic equation on divergence-free vector fields.
• Employs the Duhamel formula to reformulate the Navier-Stokes equation as a fixed-point problem in a suitable Banach space of functions.
• Applies the Gagliardo-Nirenberg and Young inequalities to control nonlinear terms in the energy estimates.
• Uses the energy equality and Gronwall's inequality to prove uniqueness of Leray weak solutions in two dimensions.

Experimental results

Research questions

• RQ1 Can dispersive and smoothing estimates for the Bochner Laplacian be established on non-compact Riemannian manifolds with negative Ricci curvature?
• RQ2 Can these estimates be used to prove global well-posedness for the incompressible Navier-Stokes equations on such manifolds?
• RQ3 Is the uniqueness of Leray weak solutions preserved in two dimensions on non-compact manifolds with negative curvature?
• RQ4 How does the structure of the nonlinearity and curvature affect the regularity and uniqueness of solutions?
• RQ5 What role does the pressure term play in the energy equality and uniqueness proof on curved manifolds?

Key findings

• Dispersive and smoothing estimates are established for the Bochner Laplacian on non-compact manifolds with negative Ricci curvature, including hyperbolic spaces.
• These estimates enable the proof of Fujita-Kato-type global well-posedness results for the incompressible Navier-Stokes equations on such manifolds.
• The energy equality is rigorously derived for weak solutions, ensuring conservation of energy and divergence-free evolution.
• Uniqueness of Leray weak solutions is proven in two dimensions on non-compact manifolds via Gronwall's inequality applied to the difference of two solutions.
• The counterexample in [32] for non-uniqueness on H² is excluded under the proposed definition of weak solution due to the requirement that pressure be in L²_T L².
• The method ensures that initial data in critical or subcritical spaces lead to global solutions with uniform control via energy and nonlinear estimates.