On Optimal Estimates for the Laplace-Leray Commutator in Planar Domains with Corners

TL;DR

This paper investigates the behavior of the Laplace-Leray commutator in planar domains with corners, revealing that strict control estimates valid in smooth domains fail in general but can be recovered in cone-shaped domains using weighted norms.

Contribution

It demonstrates the failure of strict commutator control in domains with corners and identifies conditions under which control can be restored in cone-shaped domains.

Findings

Strict control fails in domains with corners.

Control can be recovered in cone domains with weighted norms.

Results impact the analysis of Navier-Stokes equations in non-smooth domains.

Abstract

For smooth domains, Liu et al. (Comm. Pure Appl. Math. 60: 1443-1487, 2007) used optimal estimates for the commutator of the Laplacian and the Leray projection operator to establish well-posedness of an extended Navier-Stokes dynamics. In their work, the pressure is not determined by incompressibility, but rather by a certain formula involving the Laplace-Leray commutator. A key estimate of Liu et al. controls the commutator strictly by the Laplacian in energy norm at leading order. In this paper we show that this strict control fails in a large family of bounded planar domains with corners. However, when the domain is an infinite cone, we find that strict control may be recovered in certain power-law weighted norms.