Liouville theorems for the Stokes equations with applications to large time estimates

TL;DR

This paper establishes Liouville theorems for non-stationary Stokes equations in exterior domains, leading to large time estimates and properties of the Stokes semigroup, with implications for fluid dynamics in unbounded regions.

Contribution

It proves Liouville theorems under decay conditions and demonstrates that the Stokes semigroup is bounded and analytic on certain function spaces, extending understanding of fluid flow behavior.

Findings

Stokes semigroup is a bounded analytic semigroup on L^{ty}_{msigma} for n  3

Large time estimates are obtained for n=2 with zero net force

Liouville theorems are established for the non-stationary Stokes equations in exterior domains

Abstract

We study Liouville theorems for the non-stationary Stokes equations in exterior domains in $ \mathbb{R}^{n}$ under decay conditions for spatial variables. As applications, we prove that the Stokes semigroup is a bounded analytic semigroup on $L^{ \infty}_{ \sigma}$ of angle $ \pi/2$ for $n \geq 3$. We also prove large time estimates for $n=2$ with zero net force.