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

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.