  We are concerned with a family of dissipative active scalar equation with
velocity fields coupled via multiplier operators that can be of high-order. We
consider sub-critical values for the fractional diffusion and prove global
well-posedness of solutions with small initial data belonging to a framework
based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the
smallness condition is with respect to the weak norm of this space, some
initial data with large $L^{2}$-norm can be considered. Self-similar solutions
are obtained depending on the homogeneity of the initial data and couplings.
Also, we show that solutions are asymptotically self-similar at infinity. Our
results can be applied in a unified way for a number of active scalar PDEs like
1D models on dislocation dynamics in crystals, Burguer's equations, 2D
vorticity equation, 2D generalized SQG, 3D magneto geostrophic equations, among
others.
