Global solutions for a family of GSQG front equations

We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<\alpha <2$, where $\alpha =1$ corresponds to the SQG equation and $\alpha =2$ corresponds to the incompressible Euler equations. This result completes previous global well-posedness results for $0<\alpha \le 1$. We also use contour dynamics to derive the GSQG front equations for $1<\alpha <2$.