Sharp L^p estimates for singular transport equations

TL;DR

This paper establishes sharp $L^p$ estimates for a singular transport equation involving the Hilbert transform, introduces a cascading solution method, and demonstrates sharp growth in specific incompressible velocity fields, with applications in fluid dynamics and general relativity.

Contribution

It introduces a novel cascading solution approach for singular transport equations and proves an invariance property of the Hilbert transform, advancing understanding of $L^p$ behavior in such equations.

Findings

Proved sharp $L^p$ estimates for the singular transport equation.

Established an invariance result for the Hilbert transform.

Provided an example of $L^p$ growth in an incompressible velocity field.

Abstract

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.