Archiv der Mathematik ( IF 0.6 ) Pub Date : 2023-10-11 , DOI: 10.1007/s00013-023-01923-5 Thieu Huy Nguyen , Thi Ngoc Ha Vu
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.
中文翻译：
爱因斯坦流形上的纳维-斯托克斯方程的里奇曲率和初始数据大小
考虑一个非紧爱因斯坦流形 ( M , g )，其具有负 Ricci 曲率张量（曲率常数\(r<0\ ) 的 \({\textrm{Ric}}_{ij}=rg_{ij}\) ）。用\(\Gamma (TM)\)表示M上所有向量场的集合，我们研究纳维-斯托克斯方程
$$\begin{对齐} {\left\{ \begin{array}{ll} \partial _t u + \nabla _u u + {\text {grad}}\pi = {\text {div}}(\nabla u + \nabla u^t)^{\sharp },\, {\text {div}}u=0,\\ u|_{t=0}= u_0 \in \Gamma (TM), {\text {div}}u_0=0，\end{array}\right。} \end{对齐}$$对于向量场\(u\in \Gamma (TM)\)。给定任何初始数据\(u_0\in \Gamma (TM)\)，我们证明如果曲率常数r足够大，那么爱因斯坦流形 ( M , g ) 上的纳维-斯托克斯方程总是有唯一解\ (u(\cdot ,t)\in \Gamma (TM)\)是为所有\(t\ge 0\)和\(u(\cdot ,0)=u_0\)定义的。我们还证明了在适当条件下解的指数衰减。
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