On the Second Iterate for Critically Diffusive Active Scalar Equations

We consider an iterative resolution scheme for a class of active scalar equations with a fractional power γ of the Laplacian and focus our attention on the second iterate. In the case of critical diffusivity, we extract information relevant to Well-posedness questions in scale-invariant spaces. Our results are Two-fold: we prove continuity of the bilinear operator in ${\dot{B}^{0}_{\infty,1}}$ ; for equations with an even symbol we show that the ${B^{-1/2}_{\infty,q}}$ -regularity, where q > 2, is in a sense a minimal necessary requirement on the solution.