Ricci curvature and the size of initial data for the Navier–Stokes equations on Einstein manifolds

Abstract

Consider a noncompact Einstein manifold (M, g) with negative Ricci curvature tensor (${\textrm{Ric}}_{ij}=rg_{ij}$ for a curvature constant $r<0$). Denoting by $\Gamma (TM)$ the set of all vector fields on M, we study the Navier–Stokes equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u + \nabla _u u + {\text {grad}}\pi = {\text {div}}(\nabla u + \nabla u^t)^{\sharp },\, {\text {div}}u=0,\\ u|_{t=0}= u_0 \in \Gamma (TM), {\text {div}}u_0=0, \end{array}\right. } \end{aligned}$$

for the vector field $u\in \Gamma (TM)$. Given any initial datum $u_0\in \Gamma (TM)$, we prove that if the curvature constant r is large enough, then the Navier–Stokes equations on the Einstein manifold (M, g) always have a unique solution $u(\cdot ,t)\in \Gamma (TM)$ which is defined for all $t\ge 0$ with $u(\cdot ,0)=u_0$. We also prove the exponential decay of solutions under appropriate conditions.