Existence and Stability of Steady-State Solutions with Finite Energy for the Navier-Stokes equation in the Whole Space

TL;DR

This paper proves the existence, uniqueness, and stability of finite-energy steady-state solutions to the Navier-Stokes equations in the entire space under specific forcing conditions, advancing understanding of fluid behavior in unbounded domains.

Contribution

It introduces new conditions on the forcing function that guarantee finite-energy solutions and demonstrates their stability and uniqueness in the whole space setting.

Findings

Existence of solutions with finite Dirichlet integral under certain forcing.

Construction of finite-energy solutions when forcing has no low modes and small ratio to viscosity.

Proven stability of these solutions against perturbations with finite initial energy.

Abstract

We consider the steady-state Navier-Stokes equation in the whole space $\mathbb{R}^3$ driven by a forcing function $f$. The class of source functions $f$ under consideration yield the existence of at least one solution with finite Dirichlet integral ($\|\nabla U\|_2<\infty$). Under the additional assumptions that $f$ is absent of low modes and the ratio of $f$ to viscosity is sufficiently small in a natural norm we construct solutions which have finite energy (finite $L^2$ norm). These solutions are unique among all solutions with finite energy and finite Dirichlet integral. The constructed solutions are also shown to be stable in the following sense: If $U$ is such a solution then any viscous, incompressible flow in the whole space, driven by $f$ and starting with finite energy, will return to $U$.