Existence and regularity of co-rotating and traveling-wave vortex solutions for the generalized SQG equation

By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and traveling-wave vortex solutions for the gSQG equation, which extends the result of Hmidi and Mateu [28] to α ∈ [ 1 , 2 ) . Moreover, we obtain the C ∞ regularity of vortices boundary and the convexity of each vortices component.