Well-posedness and global in time behavior for $L^p$-mild solutions to the Navier-Stokes equation on the hyperbolic space

Abstract

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 $\mathbb{H}^n$.