Finite time blowup of solutions of the hydrostatic Euler equations

In this paper we study the smooth solutions of the hydrostatic Euler equations. By a careful construction we establish an analytical lemma; in this lemma, we find the explicit formula in a specific class of functions that maximizes the \(L^2\) $L^2$ integral. Basing on a classical invariant transformation we simplify the problem. As a result, for a more general class of initial data we obtain the finite time blowup of the smooth solutions of the hydrostatic Euler equations in a strip domain by this lemma and repeatedly apply the method of characteristics.