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The Boltzmann equation without Grad’s angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially… (More)

- Chao-Jiang Xu, Claude Zuily
- 1997

We shall consider weak solutions of (1.1) i.e. solutions belonging to the space M 1(Ω) = { u ∈ L2(Ω,RN ) : Xi u ∈ L2(Ω,RN ), i = 1, . . . ,m } . A particular case of our main result will be the following Theorem 1.1. Assume that the data aij and f α are C∞ functions o f their arguments and that f α satisfies (1.2). Then every weak solution u ∈ M 1(Ω) of… (More)

- Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu, YOSHINORI MORIMOTO
- 2013

We prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for… (More)

This is a continuation of our series of works for the inhomogeneous Boltzmann equation. We study qualitative properties of classical solutions, precisely, the full regularization in all variables, uniqueness, non-negativity and convergence rate to the equilibrium. Together with the results of Parts I and II about the well posedness of the Cauchy problem… (More)

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat… (More)

There are many papers concerning the propagation of regularity for the solution of the Boltzmann equation (cf. [5, 6, 8, 9, 13] and references therein). In these works, it has been shown that the Sobolev or Lebesgue regularity satisfied by the initial datum is propagated along the time variable. The solutions having the Gevrey regularity for a finite time… (More)

- HUA CHEN, CHAO-JIANG XU
- 2009

This is a non-linear diffusion equation, and the coefficients āi j, c̄ depend on the solution f . Here we are mainly concerned with the Gevrey class regularity for the solution of the Landau equation. This equation is obtained as a limit of the Boltzmann equation when the collisions become grazing (see [8] and references therein). Recently, a lot of… (More)

- Hua Chen, Weixi Li, Chao-Jiang Xu, HUA CHEN
- 2017

In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.

- Wei-Xi Li, Di Wu, Chao-Jiang Xu
- SIAM J. Math. Analysis
- 2016