Shock formation for the 2D rotating shallow water equations with non-zero vorticity

TL;DR

This paper establishes the conditions under which shocks form in the 2D rotating shallow water equations with non-zero vorticity, providing a detailed geometric description of the shock formation mechanism.

Contribution

It constructs initial data leading to finite-time blow-up and describes the shock formation process with non-zero vorticity, a novel analysis for this system.

Findings

Finite-time blow-up for solutions with large vorticity

Shock formation characterized by collapse of characteristic hypersurfaces

First derivatives blow up while potential vorticity remains Lipschitz

Abstract

In the paper, the shock formation for the two-dimensional rotating shallow water system is established. We construct a large class of initial data which leads to the finite-time blow-up for the solutions. Moreover, the solutions are allowed to have non-zero large vorticity (in derivative sense), even up to the shock. Our results provide the first complete geometric description of the shock formation mechanism to the two-dimensional rotating shallow water system with vorticity. The formation of shock is characterized by the collapse of the characteristic hypersurfaces, where the first-order derivatives of the velocity, the height, and the specific vorticity blow up while the potential vorticity remains Lipschitz continuous. The methods developed in this paper should also be useful in studying the shock formation for the Euler equations with various source terms and a class of quasilinear Klein-Gordon equations in multi-dimensions.