Existence, Uniqueness, and Localization Results for a Dirichlet Problem in Geophysics

Abstract In the present paper, we investigate the existence, uniqueness, and localization of solutions for a recently derived model for gyres, formulated as a Dirichlet problem. This problem is studied from two perspectives. In the first part, we establish the existence and uniqueness of a weak solution by means of the Minty–Browder theorem. For the nonlinear term, we assume a suitable monotonicity condition together with an exponential growth condition. Here, the growth condition is admissible since the problem is posed in dimension two, which allows us to use the Trudinger–Moser inequality. In the second part, we prove the existence of a localized positive solution within a conical annular region by means of the Moser–Harnack inequality. This approach also yields multiplicity results when the required conditions are satisfied on disjoint sets. From a physical perspective, an upper bound for the solution reflects the dynamical intensity of the flow, while a lower bound indicates the persistence of ocean circulation in a given region. These bounds are determined by the behavior of the vorticity function.