In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.
We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space L 2 v (B 3/2 2,1 ), then the Cauchy problem of Landau equation admits a global solution belongs to L ∞ t L 2 v (B 3/2 2,1 ). The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.2010 Mathematics Subject Classification. Primary: 35H20, 35E15; Secondary:76P05, 82C40.
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