On the collapse of the local Rayleigh condition and the finite time blow-up for the semi-lagrangian equations

TL;DR

This paper investigates the conditions under which solutions to certain 2D hydrostatic Euler and semi-lagrangian equations develop singularities or blow up in finite time, highlighting the collapse of the local Rayleigh condition.

Contribution

It demonstrates the finite time blow-up of solutions to semi-lagrangian equations and identifies conditions leading to the collapse of the local Rayleigh condition.

Findings

Solutions can develop singularities under certain assumptions.

Finite time blow-up occurs for solutions to semi-lagrangian equations.

Necessary conditions for global solvability are identified.

Abstract

In this paper we study the propagation of the local Rayleigh condition for the two-dimensional hydrostatic Euler equation in the framework of the local well-posedness result by Masmoudi and Wong \cite{MaTKW12}. We show under certain assumptions that such solutions will develop singularities or collapse the local Rayleigh condition. In addition, we find necessary conditions for the global solvability. Finally, we establish the finite time blow-up of solutions to the semi-lagrangian equations introduced by Brenier in \cite{Bre99} for certain class of initial data.