A family of dissipative active scalar equations with singular velocity and measure initial data

This paper considers a family of generalized active scalar equations, with fractional dissipation, whose velocity fields are more singular than Riesz transform. We prove global well-posedness results for small initial data belonging to Besov–Morrey spaces, which contain strongly singular functions and measures concentrated at points (Diracs) and on smooth curves. Self-similar solutions are obtained for initial data and coupling-velocity operator with correct homogeneities. We also show an asymptotic behavior result and obtain a class of asymptotically self-similar solutions.