Gevrey regularity for solutions of the non-cutoff Boltzmann equation: The spatially inhomogeneous case

In this paper we consider the non-cutoff Boltzmann equation in the spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order 1/(2s)1/(2s) in the velocity variable vv and order 11 in the space variable xx. The strategy relies on our recent results for the spatially homogeneous case [T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff, J. Differential Equations 253 (4) (2012), 1172–1190. http://dx.doi.org/10.1016/j.jde.2012.04.023]. Rather, we need much more intricate analysis additionally in order to handle with the coupling of the double variables. Combining with the previous result mentioned above, it gives a characterization of the Gevrey regularity of the particular kind of solutions to the non-cutoff Boltzmann.