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Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity

DOI: 10.1155/2014/584169

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The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of the solutions in non-Maxwellian and strong singularity cases. 1. Introduction The standard form of the initial value problem for the spatially homogeneous nonlinear noncutoff Boltzmann equation is expressed as follows: where is a fixed positive number and denotes the density distribution function for velocity at time . The Boltzmann collision operator is expressed as follows: where is the unit sphere of . For , The Boltzmann collision cross section is a function that was assumed to be the following form: where the kinetic factor . The angular part has a singularity that satisfies for constant and : Cases , are considered mild singularity and strong singularity, respectively. The following norms of weighted function spaces are introduced: where . is the corresponding pseudo-differential operator. The definition of the Gevrey space can now be listed; compare [1–5]. Definition 1. For , the smooth function which is the Gevrey space with index if there exists a positive constant such that, for any , or, equivalently, where It is indicated that is also equivalent to the fact that there exists such that . Research on the Gevrey regularity of the Boltzmann equation can be traced back to the work of Ukai [6], who constructed a unique local solution in Gevrey space for both spatially homogeneous and inhomogeneous noncutoff Boltzmann equations. In 2004, Desvillettes and Wennberg [7] gave a conjecture of the Gevrey smoothing effect. Five years later, the propagation of Gevrey regularity for solutions of the nonlinear spatially homogeneous Boltzmann equation with Maxwellian molecules is obtained in [8]. In that same year, Morimoto et al. [4] studied linearized cases and proved the Gevrey regularity of solutions without any extra assumption for the initial datum. They then considered the solutions with Maxwellian decay in [9]; that is, a positive number exists such that, for any , Under the hypotheses of , , , and the modified kinetic factor , they showed the Gevrey smooth property for this type of solutions to the Cauchy problem of the nonlinear homogeneous Boltzmann equation. By using the original definition of kinetic factor, Zhang and Yin [10] extended the above result in a general framework: and . In this paper, the same issue in the strong singularity case is disussed. To discuss this issue properly, some notations are introduced. For any , and , the following expression is denoted: For

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