Global solutions of a surface quasigeostrophic front equation

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on R that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.