The authors prove that small L^3 initial data yield unique global mild solutions with exponential decay for the Navier-Stokes equations on 3-manifolds satisfying -b² ≤ K ≤ -a² < 0.
Balentine, Well-posedness and global in time behavior forL p-mild solutions to the Navier-Stokes equation on the hyperbolic space, arXiv preprint arXiv:2008.01850, 2020
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Exponential stability for the three-dimensional Navier-Stokes equations on negatively curved manifolds
The authors prove that small L^3 initial data yield unique global mild solutions with exponential decay for the Navier-Stokes equations on 3-manifolds satisfying -b² ≤ K ≤ -a² < 0.