Weak solutions for generalized large-scale semigeostrophic equations

Abstract

We prove existence, uniqueness and continuous dependence on initialdata of global weak solutions to the generalized large-scalesemigeostrophic equations with periodic boundary conditions. Thisfamily of Hamiltonian balance models for rapidly rotating shallowwater includes the $L_1$ model derived by R. Salmon in 1985 and its2006 generalization by the second author. The analysis is based on thevorticity formulation of the models supplemented by a nonlinearvelocity-vorticity relation. The results are fundamentally due to theconservation of potential vorticity. While classical solutions areknown to exist provided the initial potential vorticity ispositive---a condition which is already implicit in the formalderivation of balance models, we can assert the existence of weaksolutions only under the slightly stronger assumption that thepotential vorticity is bounded below by $\sqrt{5}-2$ times theequilibrium potential vorticity. The reason is that thenonlinearities in the potential vorticity inversion are felt morestrongly when working in weaker function spaces. Anothermanifestation of this effect is that point-vortex solutions are notsupported by the model even in the special case when the potentialvorticity inversion gains three derivatives in spaces of classicalfunctions.