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

Local existence and uniqueness for the two-dimensional hydrostatic Euler equations in Sobolev spaces has been established by Masmoudi and Wong (Arch Rational Mech Anal 204:231–271, 2012) under the local Rayleigh condition. Under certain assumptions, we show that such solution will either develop singularities or produce the collapse of the local Rayleigh condition. In addition, we find necessary conditions for global solvability in Sobolev spaces. Finally, for certain class of initial data, we establish the finite time blow-up of solutions of the semi-Lagrangian equations introduced by Brenier (Nonlinearity 12:495–512, 1999). Our proof relies on new monotonicity identities for the solution of the hydrostatic Euler equations under the local Rayleigh condition.