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. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

The authors acknowledge financial support from Grant PID2019-106715GBC21 funded by MCIN/AEI/ 10.13039/501100011033. J.C. Marrero and E. Padr´on acknowledge financial support from the Spanish Ministry of Science and Innovation under grant PGC2018-098265-B-C32.

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