Mahmut Calik

• We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of PV-conserving balanced models for rapidly rotating shallow water includes the $L_1$ model derived by R. Salmon in 1985 and its 2006 generalization by M. Oliver. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the 2D Euler equations. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic $L̂p$ theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation. We note that our work, besides the nonlinearity of the potential vorticity inversion operator, differs also from earlier works with respect to the fact that the velocity is not divergence-free in our case. Moreover, we prove existence, uniqueness and continuous dependence on initial data of global weak solutions. Radon-measured potential vorticities as can be shown to make sense in Euler-$\alpha$ case, however, appear to break down altogether here. Finally, as a first step towards model justification we prove that, for balanced initial data, solutions of the rotating shallow water model stay $O(\varepsilon)$-accurately close to the solution of the $L_1$ model on finite time intervals of order 1.