  We consider a family of dissipative active scalar equations outside the
$L^{2}$-space. This was introduced in [D. Chae, P. Constantin, J. Wu, to appear
in IUMJ (2014)] and its velocity fields are coupled with the active scalar via
a class of multiplier operators which morally behave as derivatives of positive
order. We prove global well-posedness and time decay of solutions, without
smallness assumptions, for initial data belonging to the critical Lebesgue
space $L^{\frac{n}{\gamma-\beta}}(\mathbb{R}^{n})$ which is a class larger than
that of the above reference. Symmetry properties of solutions are investigated
depending on the symmetry of initial data and coupling operators.

Analysis of PDEs