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

Víctor Chaves-Santos, Lucas C. F. Ferreira

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.

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

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 -norms and thus allow for certain classes of large -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.
Paper Structure (16 sections, 23 theorems, 219 equations, 2 figures)

This paper contains 16 sections, 23 theorems, 219 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the Manuscript
  3. Review of volume estimates
  4. Upper bounds
  5. Lower bounds
  6. Morrey spaces on manifolds
  7. Heat equation on non-compact manifolds
  8. Heat equation with Bochner Laplacian
  9. Dispersive estimates
  10. Smoothing estimates
  11. Riesz transform
  12. Navier-Stokes equations
  13. A general class of non-compact manifolds
  14. The modified case with viscosity
  15. Einstein manifolds with negative curvature
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $(M,g)$ be a complete Riemannian manifold of dimension $m$. If $\operatorname{Ric}\geq -K(m-1)$ for $K>0$, then there exist constants $C(m)>0$ such that for $0<c<R_{1}<R_{2}$ with $k=\sqrt{K}(m-1)$ and for $0<R_{1}<R_{2}$.

Figures (2)

  • Figure 1: The blue arrow indicates the extra assumption for the case $\mathcal{M}^g_{p,\lambda}$
  • Figure 2: The blue arrow indicates the extra assumption for the case $\mathcal{M}^{g}_{p,\lambda}$

Theorems & Definitions (33)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2: Hölder's inequality
  • Lemma 3.3
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1: Dispersive: $\mathcal{M}_{p,\lambda }^{g}\rightarrow L^{\infty }$
  • ...and 23 more