Classical Solutions for a Generalized Euler Equation in Two Dimensions

It is well known that the Euler equations in two spatial dimensions have global classical solutions. We provide a new proof which is analytic rather than geometric. It is set in an abstract framework that applies to the so-called lake and the great lake equations describing weakly non-hydrostatic effects of bottom topography on the motion of shallow water. The key ingredient is a new L p estimate on the nonlinear term. The estimate is used to develop a global Hm theory for bounded domains in R2 which is similar in spirit to a 1975 paper by R. Temam. It also provides explicit bounds on the Hm norm which grow like exp exp t..