Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces
Solution to the Boltzmann equation in velocity-weighted Chemin-Lerner type spaces
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DOI：
10.3934/krm.2018051
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发表时间：
2017-09
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影响因子：
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通讯作者：
Renjun Duan;Shota Sakamoto
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文献类型：
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作者：
Renjun Duan;Shota Sakamoto
本文研究d维全空间中整体Maxwell附近的Boltzmann方程.在Chemin-Lerner型空间中建立了方程Cauchy问题关于相变量$（x，v）$的唯一的整体时间温和解。硬和软势角截止被认为是。引入新的整体适定性函数空间，从本质上处理了软势的情形，其关键是速度变量取加权上确界范数，空间变量在s阶Besov空间中，其中包含空间临界正则性.证明的基础上的时间衰减性质的解决方案的线性化方程与自举参数。特别地，在硬势情况下的线性分析是由于半群理论，其中额外的时间衰减在处理关于空间变量的L^2 $中的初始数据中起作用。在软势情形下，对于线性方程组的时间衰减，我们借用了纯能量方法的结果，并通过L^2 $-L ^infty$相互作用技巧将其推广到L^infty框架下.与硬势相反，为了得到非线性柯西问题的整体解，软势在x$中的L^1 $可积性是必要的。
In this paper we study the Boltzmann equation near global Maxwellians 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 $s\geq d/2$ 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 space variable. In case of soft potentials, for the time-decay of linear equations we borrow the results basing on the pure energy method and further extend them to those in $L^\infty$ framework through the technique of $L^2$--$L^\infty$ 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.