Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath
Exponential trend to equilibrium for the inelastic Boltzmann equation driven by a particle bath
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粒子浴驱动的非弹性玻尔兹曼方程达到平衡的指数趋势
DOI：
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发表时间：
2015
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通讯作者：
B. Lods
中科院分区：
文献类型：
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作者：
J. Cañizo;B. Lods
本文研究了非弹性硬球（具有常数恢复系数α∈（0，1））在具有固定Maxwell分布的主介质诱导的热化作用下的空间齐次Boltzmann方程.我们证明了相关的初始值问题的解决方案，指数快速收敛到唯一的平衡解。该证明结合了对线性化半群的仔细谱分析以及熵估计。在α接近1的弱非弹性区域中，趋于平衡的趋势保持不变，并且收敛速度是明确的，仅取决于弹性线性碰撞算子的谱隙。
We consider the spatially homogeneous Boltzmann equation for inelastic hard spheres (with constant restitution coefficient α∈(0,1)) under the thermalization induced by a host medium with a fixed Maxwellian distribution. We prove that the solution to the associated initial-value problem converges exponentially fast towards the unique equilibrium solution. The proof combines a careful spectral analysis of the linearised semigroup as well as entropy estimates. The trend towards equilibrium holds in the weakly inelastic regime in which α is close to 1, and the rate of convergence is explicit and depends solely on the spectral gap of the elastic linear collision operator.