论文标题
一维非弹性Boltzmann方程：规律性\＆唯一相似的型号的唯一性
One-dimensional inelastic Boltzmann equation: Regularity \& uniqueness of self-similar profiles for moderately hard potentials
论文作者
论文摘要
我们证明了具有适度硬势的一维非弹性玻尔兹曼方程的自相似曲线的独特性，即形式的碰撞内核| $ \ bullet $ | $γ$ for $γ$> 0小（明确量化）。我们的结果为无弹力玻尔兹曼模型的自相似曲线提供了第一个唯一性语句，除了明确解决的麦克斯韦相互作用情况（对应于$γ$ = 0）之外，还允许强烈的非弹性。我们的方法依赖于对相应的麦克斯韦模型的扰动参数，通过仔细研究相关的线性化操作员。特别是，本文的一部分专门用于在适当的加权Sobolev空间中麦克斯韦模型的趋势，这是已知在较弱的拓扑中保持的结果的扩展。我们的结果可以看作是在一维环境中朝着Ernst \＆Brito（2002）中的一个猜想的全面证明的第一步，该结果涉及确定解决方案的长期行为。
We prove uniqueness of self-similar profiles for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, that is with collision kernel of the form | $\bullet$ | $γ$ for $γ$ > 0 small enough (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $γ$ = 0). Our approach relies on a perturbation argument from the corresponding Maxwell model through a careful study of the associated linearised operator. In particular, a part of the paper is devoted to the trend to equilibrium for the Maxwell model in suitable weighted Sobolev spaces, an extension of results which are known to hold in weaker topologies. Our results can be seen as a first step towards a full proof, in the one-dimensional setting, of a conjecture in Ernst \& Brito (2002) regarding the determination of the long-time behaviour of solutions to inelastic Boltzmann equation.
