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

Braden Balentine

Published 2020-08-04Version 1

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 $L^p$ initial data in the range $1 < p < \infty$, global in time results for small initial data, and $L^p$ norm decay results for both $u$ and $\nabla u$. As part of this, we extend to the hyperbolic space $\mathbb{H}^n$ 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 theory for mild solutions on $\mathbb{H}^n$. While the statements of the results are the same as in the Euclidean case, the methods of the proofs are at times different.