Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations

Abstract

We prove existence and uniqueness of global classical 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 L1 model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. 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 Lp 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.