Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

Renjun Duan; Shota Sakamoto

doi:10.3934/krm.2018051

Article Contents

2018, Volume 11, Issue 6: 1301-1331. Doi: 10.3934/krm.2018051

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Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces

Renjun Duan^1, , and
Shota Sakamoto^2,

1.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
2.
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan

* Corresponding author: Renjun Duan
* Corresponding author: Renjun Duan

Received: September 30, 2017

Published: June 2018

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Abstract

In this paper the Boltzmann equation near global Maxwellians is studied in the $d$ -dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect to the phase variable $(x,v)$ . Both hard and soft potentials with angular cutoff are considered. The new function space for global well-posedness is introduced to essentially treat the case of soft potentials, and the key point is that the velocity variable is taken in the weighted supremum norm, and the space variable is in the $s$ -order Besov space with $s≥ d/2$ including the spatially critical regularity. The proof is based on the time-decay properties of solutions to the linearized equation together with the bootstrap argument. Particularly, the linear analysis in case of hard potentials is due to the semigroup theory, where the extra time-decay plays a role in coping with initial data in $L^2$ with respect to the space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results based on the pure energy method and further extend them to those in $L^∞$ framework through the technique of $L^2-L^∞$ interplay. In contrast to hard potentials, $L^1$ integrability in $x$ of initial data is necessary for soft potentials in order to obtain global solutions to the nonlinear Cauchy problem.

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Mathematics Subject Classification: Primary: 35Q20; Secondary: 76P05, 82B40.

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