Existence of strong solutions to the generalized inverse of the quasi-geostrophic equations

Existence of strong (i.e. classical) solutions to the generalized inverse of the threedimensional quasi-geostrophic equations, describing the large-scale motion of the atmosphere and oceans, is proved for a finite time interval. Both the dissipative and the nondissipative cases are considered. The spatial domain considered is doubly periodic in the horizontal directions and is bounded above and below by rigid, horizontal surfaces. The generalized inverse is defined as the solution to the Euler–Lagrange equations, obtained by minimizing a weighted sum of errors in the quasi-geostrophic equations, boundary conditions, and data, and thus represents a solution that approximates both the equations and the data available inside the domain. The proof relies on the Schauder fixed-point theorem applied to the appropriate Hölder function spaces. The finite time interval over which the proof is valid is not arbitrary, but depends on the norms of the initial conditions in such a way that, as the norms of the initial conditions increase, the time interval decreases.