We consider the Euler equations of an incompressible homogeneous fluid in a
thin two-dimensional layer -infinity < x < +infinity, 0 < z < epsilon, wit
h slip boundary conditions at z = 0, z = epsilon and periodic boundary cond
itions in x. After rescaling the vertical variable and letting epsilon go t
o 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 bo
undary conditions in x. We show that the corresponding initial-value proble
m is locally, but generally not globally, solvable in the class of smooth s
olutions with strictly convex horizontal velocity profiles, with constant s
lopes at z = 0 and z = 1.