The viscosity limit of fluid flows with growth/decay conditions at infinity

TL;DR

This paper establishes the well-posedness of the Navier-Stokes equations for vector fields with specific growth conditions at infinity in certain function spaces, and analyzes the viscosity limit as it approaches zero.

Contribution

It introduces new function space frameworks and analytical techniques to handle fluid flows with unbounded growth at infinity, extending the understanding of the Navier-Stokes and Euler equations.

Findings

Solutions depend continuously on viscosity parameter  and converge to Euler solutions as  0+

Well-posedness is proved in weighted Sobolev spaces for vector fields with growth rate  < 1/2

New analytical methods include properties of conjugated heat flow and a variant of the Lie-Trotter product formula.

Abstract

We prove that the Navier-Stokes equation is well-posed in function spaces on $\mathbb{R}^d$, $d\ge 2$, that contain vector fields of order $O(|x|^\kappa)$ as $|x|\to\infty$ with $\kappa<1/2$. The corresponding solutions depend continuously on the viscosity parameter $\nu\ge 0$ and converge to the solutions of the Euler equation as $\nu\to 0+$. Our proof is based on the properties of the conjugated heat flow on weighted Sobolev spaces and on a new variant of the Lie-Trotter product formula for nonlinear semigroups.