Sharp Strong Convergence in Ideal Flows
Haroune Houamed, Marc Magaña
TL;DR
This work rigorously derives the Euler equation as the zero-inverse-Rossby-radius limit of the 2D QGSW model for Yudovich data, proving strong convergence of vorticities in $L^\infty_t L^p_x$ for all finite $p$ and, under continuity assumptions, in $L^\infty_{t,x}$. The authors introduce Extrapolation Compactness, performing a three-level program that first achieves convergence in a low-regularity Besov-type space, then reduces high-regularity convergence to the evanescence of very high frequencies, and finally controls high-frequency transport effects via refined Littlewood–Paley analysis in endpoint spaces. The endpoint convergence is shown to be sharp for vortex-patch-type initial data, with a vortex-patch rigidity argument linked to kernel monotonicity and rotating/stationary solutions. The results not only establish a precise, quantitative connection between QGSW and Euler in a critical setting, but also provide a robust methodological framework applicable to similar singular limits in active scalar equations.
Abstract
We investigate the strong convergence of weak solutions to the two-dimensional Quasi-Geostrophic Shallow-Water (QGSW) equation as the inverse Rossby radius tends to zero. In this limit, we recover the Yudovich solution of the incompressible Euler equations. We prove that the vorticity convergence holds in $L^\infty_t L^p_x$, for any finite integrability exponent $p<\infty$. This extends to the case $p=\infty$ provided that the initial vorticities are continuous and converge uniformly. We also discuss the sharpness of this limit by demonstrating that the continuity assumption on the initial data is necessary for the endpoint convergence in $L^\infty_{t,x}$. The proof of the strong convergence relies on the {\em Extrapolation Compactness} method, recently introduced by Arsénio and the first author to address similar stability questions for the Euler equations. The approach begins with establishing the convergence in a lower regularity space, at first. Then, in a later step, the convergence to Yudovich's vorticity of Euler equations in Lebesgue spaces comes as a consequence of a careful analysis of the evanescence of specific high Fourier modes of the QGSW vorticity. A central challenge arises from the absence of a velocity formulation for QGSW, which we overcome by employing advanced tools from Littlewood Paley theory in endpoint settings. The sharpness of the convergence in the endpoint $L^\infty_{t,x}$ case is obtained in the context of vortex patches, drawing insights from key findings on uniformly rotating and stationary solutions of active scalar equations.