# Asymptotic behavior of the steady Navier-Stokes equation on the hyperbolic plane

## TL;DR

This paper investigates the long-term behavior of solutions to the steady Navier-Stokes equations in the hyperbolic plane, demonstrating decay properties of velocity, vorticity, and pressure at infinity.

## Contribution

It establishes the decay rates and asymptotic behavior of solutions to the stationary Navier-Stokes equations in the hyperbolic plane, extending understanding to non-Euclidean geometries.

## Key findings

- Velocity decays to zero at infinity
- Vorticity decay rate characterized
- Pressure behavior at infinity analyzed

## Abstract

We develop the asymptotic behavior for the solutions to the stationary Navier-Stokes equation in the exterior domain of the 2D hyperbolic space. More precisely, given the finite Dirichlet norm of the velocity, we show the velocity decays to $0$ at infinity. We also address the decay rate for the vorticity and the behavior of the pressure.