On basic velocity estimates for the plane steady-state Navier-Stokes system and its applications

TL;DR

This paper introduces new velocity estimates for steady-state Navier-Stokes solutions in convex plane domains, providing bounds on mean velocity differences based on the Dirichlet integral, with various applications discussed.

Contribution

It establishes universal bounds for velocity differences in convex domains, linking mean values to the Dirichlet integral, advancing understanding of steady Navier-Stokes solutions.

Findings

Velocity difference bounds are proportional to the square root of the Dirichlet integral.

The bounds are universal, independent of circle radii ratios.

Applications include improved estimates for flow behavior in convex domains.

Abstract

We consider some new estimates for general steady Navier-Stokes solutions in plane domains. According to our main result, if the domain is convex, then the difference between mean values of the velocity over two concentric circles is bounded (up to a constant factor) by the square-root of the Dirichlet integral in the annulus between the circles. The constant factor in this inequality is universal and does not depend on the ratio of the circle radii. Several applications of these formulas are discussed.