Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation

Léo Glangetas; Hao-Guang  Li; Chao-Jiang Xu

doi:10.3934/krm.2016.9.299

Article Contents

2016, Volume 9, Issue 2: 299-371. Doi: 10.3934/krm.2016.9.299

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Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation

1.
Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l'Université, BP.12, 76801 Saint Etienne du Rouvray
2.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072
3.
Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray

Received: November 30, 2014

Revised: September 30, 2015

Published: May 2016

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Abstract

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov's regularizing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of the fractional exponent is exactly the same as the singular index of the non-cutoff collisional kernel of the Boltzmann equation. Therefore, we get the sharp regularity of solutions in the Gevrey class and also the sharp decay of solutions with an exponential weight. We also give a method to construct the solution of the Boltzmann equation by solving an infinite system of ordinary differential equations. The key tool is the spectral decomposition of linear and non-linear Boltzmann operators.

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Mathematics Subject Classification: 35Q20, 35B65.

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