The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction

In continuation with earlier works on the shallow water equations in a rectangle [10, 11], we investigate in this article the fully inviscid nonlinear shallow water equations in space dimension two in a rectangle (0,1)ₓ × (0,1)y. We address in this article the subcritical case, corresponding to the condition (3) below. Assuming space periodicity in the y-direction, we propose the boundary conditions for the x-direction which are suited for the subcritical case and develop, for this problem, results of existence, uniqueness and regularity of solutions locally in time for the corresponding initial and boundary value problem.