Sharp Lp estimates for singular transport equations

We prove that Lp estimates for a singular transport equation are sharp by building what we call a cascading solution; the equation we consider 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 incompressible velocity field u for which the equation:∂tf+u⋅∇f=H(f) develops sharp Lp growth. The equations we study are relevant, as models, in the study of fluid equations as well as in general relativity.