Critical threshold for global regularity of Euler-Monge-Amp\`ere system with radial symmetry

TL;DR

This paper establishes a precise initial condition threshold ensuring the global regularity of the Euler-Monge-Ampère system with radial symmetry, using spectral and geometric methods, and extends results to 2D with swirl.

Contribution

It provides the first explicit critical threshold for global regularity of the EMA system with radial symmetry, employing two independent analytical approaches.

Findings

Explicit critical threshold for initial data guaranteeing global regularity.

Extension of results to 2D EMA with swirl.

Validation of threshold via spectral and geometric methods.

Abstract

We study the global wellposedness of the Euler-Monge-Amp\`ere (EMA) system. We obtain a sharp, explicit critical threshold in the space of initial configurations which guarantees the global regularity of EMA system with radially symmetric initial data. The result is obtained using two independent approaches -- one using spectral dynamics of Liu & Tadmor [Comm. Math. Physics 228(3):435-466, 2002] and another based on the geometric approach of Brenier & Loeper [Geom. Funct. Analysis 14(6):1182--1218, 2004]. The results are extended to 2D radial EMA with swirl.