Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates |
| |
Authors: | Nikolaos Bournaveas Vincent Calvez Susana Gutiérrez |
| |
Institution: | 1. University of Edinburgh, School of Mathematics, JCMB, King's Buildings , Edinburgh, UK;2. école Normale Supérieure, Département de Mathématiques et Applications , Paris, France;3. University of Nottingham, School of Mathematical Sciences, Division of Theoretical Mechanics, University of Nottingham , University Park, Nottingham, UK |
| |
Abstract: | 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. (2004
Chalub , F. A. C. C. ,
Markowich , P. A. ,
Perthame , B. ,
Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .Crossref], Web of Science ®] , Google Scholar]), Hwang et al. (2005a
Hwang , H. J. ,
Kang , K. ,
Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . Google Scholar] b
Hwang , H. J. ,
Kang , K. ,
Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . Google Scholar]) using direct and strong dispersive effects. Here, we use the weak dispersion estimates of Castella and Perthame (1996
Castella , F. ,
Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . Google Scholar]) 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. |
| |
Keywords: | Chemotaxis Dispersion estimates Kinetic equations Strichartz estimates |
|
|