---
abstract: 'We study the range of validity of the Strichartz estimates for the kinetic transport equation. We are able to give the full range in spatial dimension $n=1$ while in higher dimensions we leave open some inhomogeneous estimates. Our results extend the previous works by Castella and Perthame [@CP], Keel and Tao [@KT] and Guo and Peng [@GP]. Based on a geometric interpretation of the kinetic transport propagator we construct counterexamples over Besicovitch sets to various endpoint Strichartz estimates containing $L^\infty$-norms. We also give an application to a nonlinear kinetic model of bacterial chemotaxis where we extend the result of Bournaveas et al. [@B].'
address: 'University of Edinburgh, School of Mathematics, JCMB, King’s Buildings, Edinburgh EH9 3JZ, UK'
author:
- Evgeni Ovcharov
bibliography:
- 'KTS-2.bib'
date: 25 March 2009
nocite: '[@HKS]'
title: Strichartz Estimates for the Kinetic Transport Equation
---

[^1]

Introduction
============

Historical Remarks
------------------

In this paper we prove Strichartz estimates for the kinetic transport (KT) equation $$\label{eq: KT}
 {\partial_t u(t,x,v) + v\cdot\nabla_x u(t,x,v)} = F\coor, \quad \coor \in {\mathbb R}\times{\mathbb R}^n\times {\mathbb R}^n,$$ to which we prescribe initial data $$\label{eq: KTf}
u(0,x,v) = f(x,v).$$

Strichartz estimates for the KT equation were first proved by Castella and 