The authors prove the local existence of strong solutions of the primitive equations of the ocean with initial data in the Sobolev space . These equations are obtained from the Navier-Stokes equations by an asymptotic process where the small parameter is the ratio of the vertical scale over the horizontal scale. The domain of the ocean is assumed to be a Lipschitz-continuous domain
where , is an open set and is the depth function. The boundary of is the union of , , and , where is the boundary of . The velocity of the fluid and the pressure satisfy
where and are the dimensional nabla and Laplacian operators respectively. The novel difficulty in this model, compared with the Navier-Stokes equations, is the lack of a priori estimates on the vertical component : instead of having in (in the Navier-Stokes equations), one only has in . The authors circumvent this difficulty using anisotropic inequalities and the special structure of the nonlinear term containing to obtain the necessary estimates on the nonlinear term for obtaining the global existence of small strong solutions. In order to obtain the local existence of strong solutions for large data in the authors linearize the equations around the solution of the associated linear problem and then use the small data argument.