Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

TL;DR

This paper derives smoothing estimates for the velocity averaging operator in the kinetic transport equation within hyperbolic Sobolev spaces, characterizing exponents for boundedness and connecting to cone multiplier bounds.

Contribution

It introduces a unified approach for velocity domains and establishes new smoothing estimates, including for Besov spaces and radially symmetric data, using advanced harmonic analysis tools.

Findings

Characterization of exponents for boundedness of velocity averaging operator

Equivalence of estimates to mixed-norm bounds on cone multipliers

Enhanced smoothness results for radially symmetric initial data

Abstract

We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator $\rho$ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents $q$ and $r$, we find a characterisation of the exponents $\beta_+$ and $\beta_-$, except possibly for an endpoint case, for which $D_+^{\beta_+}D_-^{\beta_-} \rho$ is bounded from space-velocity $L^2_{x,v}$ to space-time $L^q_tL^r_x$. Here, $D_+$ and $D_-$ are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on $L^2$. We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp $\ell^p$ decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.