Vanishing viscosity limit for an expanding domain in space

TL;DR

This paper investigates the behavior of viscous incompressible flows in expanding domains as viscosity approaches zero, establishing conditions for convergence to Euler equations using energy estimates and truncation methods.

Contribution

It provides new conditions under which the vanishing viscosity limit in expanding domains converges to the full space Euler equations, extending previous results.

Findings

Established precise conditions for the vanishing viscosity limit in expanding domains.

Extended Kato's criterion to unbounded, expanding domains.

Demonstrated convergence of viscous flows to Euler solutions under these conditions.

Abstract

We study the limiting behavior of viscous incompressible flows when the fluid domain is allowed to expand as the viscosity vanishes. We describe precise conditions under which the limiting flow satisfies the full space Euler equations. The argument is based on truncation and on energy estimates, following the structure of the proof of Kato's criterion for the vanishing viscosity limit. This work complements previous work by the authors, see [Kelliher, Comm. Math. Phys. 278 (2008), 753-773] and [arXiv:0801.4935v1].