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

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 $O(\mathrm{\mid }x{\mathrm{\mid }}^{\kappa })$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ with $\kappa <1\mathrm{/}2$. 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.