WEAK SOLUTIONS FOR GENERALIZED LARGE-SCALE SEMIGEOSTROPHIC EQUATIONS

We prove existence, uniqueness and continuous dependence on initial data of global weak solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L-1 model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The analysis is based on the vorticity formulation of the models supplemented by a nonlinear velocity-vorticity relation. The results are fundamentally due to the conservation of potential vorticity. While classical solutions are known to exist provided the initial potential vorticity is positive a condition which is already implicit in the formal derivation of balance models, we can assert the existence of weak solutions only under the slightly stronger assumption that the potential vorticity is bounded below by root 5 - 2 times the equilibrium potential vorticity. The reason is that the nonlinearities in the potential vorticity inversion are felt more strongly when working in weaker function spaces. Another manifestation of this effect is that point-vortex solutions are not supported by the model even in the special case when the potential vorticity inversion gains three derivatives in spaces of classical functions.