Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

Research output: Contribution to journalArticle

Colleges, School and Institutes

External organisations

  • Department of Mathematics, Saitama University
  • Department of Mathematical Sciences, Seoul National University

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.

Details

Original languageEnglish
Pages (from-to)1-28
JournalJournal de Mathematiques Pures et Appliquees
Volume114
Early online date30 Mar 2018
Publication statusPublished - Jun 2018

Keywords

  • kinetic transport equation , averaging lemmas , hyperbolic Sobolev spaces , cone multiplier operator