  We prove sharp $L^p$ estimates for a singular transport equation by building
what we call a \emph{cascading solution}; the equation studies the combined
effect of multiplying by a bounded function and application of the Hilbert
transform. Along the way we prove an invariance result for the Hilbert
transform which could be of independent interest. Finally, we give an example
of a bounded and \emph{incompressible} velocity field for which the equation:
  $$\partial_t f +u\cdot\nabla f= H(f)$$ develops sharp $L^p$ growth. The
equations we study are relevant, as models, in the study of fluid equations as
well as in general relativity.

Analysis of PDEs