论文标题
$ l^p $ -Mild解决方案的良好的时间和全球时间行为，用于双曲线空间上的Navier-Stokes方程
Well-posedness and global in time behavior for $L^p$-mild solutions to the Navier-Stokes equation on the hyperbolic space
论文作者
论文摘要
我们在$ n $二维空间$ \ mathbb {h}^n $，$ n \ geq 2 $上研究Navier-Stokes方程的温和解决方案。我们使用Pierfelice在一类完整的Riemannian歧管上证明的色散和平滑估计值，将轻度解决方案的Fujita-Kato理论从$ \ Mathbb {r}^n $到$ \ Mathbb {h h}^n $。这包括$ l^n $初始数据的适应性结果和$ 1 <p <n $的$ l^n \ cap l^p $初始数据，小型初始数据的全球时间结果，以及$ l^n $和$ l^p $规范的时间衰减结果$ u $ $ $ u $和$ \ nabla u $。由于$ \ mathbb {h}^n $提供的其他指数时间衰减，我们能够简化$ l^n $和$ l^n $和$ l^p $ norm decay结果的证明。此外，我们能够证明$ \ mathbb {h}^n $上的温和解决方案属于比以欧几里得空间所知的更大的时空$ l^rl^q $空间，而全球解决方案的$ l^n $ norm sy a a a a a a a a a a a $ t $的$ l^n $ norm at as t $ a t $在$ \ mathbb a的$ \ mathbb { $ \ mathbb {r}^n $，$ n \ geq 3 $。作为我们工作的必要部分，我们扩展到$ \ Mathbb {h}^n $在欧几里得领域的已知事实，涉及拉普拉斯（Laplacian）产生的半群的较强的连续性和合同性。此外，我们使用光谱理论在$ \ mathbb {h}^n $的设置中为某个投影操作员建立了必要的有限性和换向属性。这项工作与Pierfelice's一起，有助于在$ \ Mathbb {H}^n $上提供完整的Fujita-Kato理论。
We study mild solutions to the Navier-Stokes equation on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $n \geq 2$. We use dispersive and smoothing estimates proved by Pierfelice on a class of complete Riemannian manifolds to extend the Fujita-Kato theory of mild solutions from $\mathbb{R}^n$ to $\mathbb{H}^n$. This includes well-posedness results for $L^n$ initial data and $L^n \cap L^p$ initial data for $1 < p < n$, global in time results for small initial data, and time decay results for the $L^n$ and $L^p$ norms of both $u$ and $\nabla u$. Due to the additional exponential time decay offered on $\mathbb{H}^n$, we are able to simplify the proofs of the $L^n$ and $L^p$ norm decay results as compared to the Euclidean setting. Additionally, we are able to show that mild solutions on $\mathbb{H}^n$ belong to a wider range of space-time $L^rL^q$ spaces than is known for Euclidean space, and that the $L^n$ norm of a global solution decays to zero as $t$ goes to infinity on $\mathbb{H}^n$, which was a question left open by Kato for $\mathbb{R}^n$, $n\geq 3$. As a necessary part of our work, we extend to $\mathbb{H}^n$ known facts in Euclidean space concerning the strong continuity and contractivity of the semigroup generated by the Laplacian. Also, we establish necessary boundedness and commutation properties for a certain projection operator in the setting of $\mathbb{H}^n$ using spectral theory. This work, together with Pierfelice's, contributes to providing a full Fujita-Kato theory on $\mathbb{H}^n$.
