Homogeneous hydrostatic flows with convex velocity profiles

We consider the Euler equations of an incompressible homogeneous fluid in a thin two-dimensional layer -infinity < x < +infinity, 0 < z < epsilon, with slip boundary conditions at z = 0, z = epsilon and periodic boundary conditions in x. After rescaling the vertical variable and letting epsilon go to zero, we get the following hydrostatic limit of the Euler equations partial derivative(l)u + u partial derivative(x)u + wa partial derivative(z)u + partial derivative(x)p = 0, (1) partial derivative(x)u + partial derivative(z)w = 0, partial derivative(z)p = 0, (2) supplemented by slip boundary conditions at z = 0 and z = 1 and periodic boundary conditions in x. We show that the corresponding initial-value problem is locally, but generally not globally, solvable in the class of smooth solutions with strictly convex horizontal velocity profiles, with constant slopes at z = 0 and z = 1.