Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of 'run and tumble'. The existence of solutions has been obtained in several papers [Chalub et al. Monatsh. Math. 142, 123--141 (2004), Hwang et al. SIAM J. Math. Anal. 36, 1177--1199 (2005)] using direct and strong dispersive effects. Here, we use the weak dispersion estimates of [Castella et al. C. R. Acad. Sci. Paris 322, 535--540 (1996)] to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.