Fast rotation limit for the 2-D non-homogeneous incompressible Euler equations

TL;DR

This paper investigates the behavior of 2D density-dependent incompressible Euler equations under rapid rotation, demonstrating convergence to a quasi-homogeneous system even with ill-prepared initial data using advanced mathematical techniques.

Contribution

It introduces a novel analysis of the fast rotation limit for density-dependent Euler equations, including cases with ill-prepared initial data, using uniform estimates and compensated compactness.

Findings

Solutions converge to quasi-homogeneous Euler system under fast rotation

Uniform estimates enable control of solutions in high regularity norms

Method handles ill-prepared initial data effectively

Abstract

In the present paper, we study the fast rotation limit for the density-dependent incompressible Euler equations in two space dimensions with the presence of the Coriolis force. In the case when the initial densities are small perturbation of a constant profile, we show the convergence of solutions towards the solutions of a quasi-homogeneous incompressible Euler system. The proof relies on a combination of uniform estimates in high regularity norms with a compensated compactness argument for passing to the limit. This technique allows us to treat the case of ill-prepared initial data.