SHARP Lp ESTIMATES FOR SINGULAR TRANSPORT EQUATIONS

Abstract. 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. 1. Background The interaction between transport and non-locality is found in a variety of physical applications such as fluid mechanics, general relativity, biological aggregation, etc. A particular equation we may consider is the 2-D Euler equations of incompressible flow: ∂tu+ (u · ∇)u = −∇p, div(u) = 0, where the pressure term, p, satisfies: ∆p = i 6=