On Liouville type theorems for the self-similar solutions to the generalized Euler equations

We study the backward discretely self-similar solutions to the generalized Euler equations, where usual Biot-Savart kernel representing the velocity in terms of the vorticity is replaced by various power of the laplacian. Under milder sufficient conditions than the previous results on the Euler equations we show that Liouville type theorems hold for the time periodic solutions to the profile equations, which means that there exists no backward discretely self-similar solutions to the generalized Euler system having such profile.