Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus
Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus
复制标题
DOI：
10.1088/0951-7715/19/4/011
复制
发表时间：
2006-04
期刊：
影响因子：
1.7
通讯作者：
C. Mouhot;L. Neumann
中科院分区：
文献类型：
--
作者：
C. Mouhot;L. Neumann
对于环面中的一类线性碰撞动力学模型，特别包括硬球的线性化玻尔兹曼方程、具有硬势和中等软势的线性化朗道方程以及半经典线性化费米子和玻色子弛豫模型，我们证明了对某些修改的索博列夫范数相关积分微分算子的显式矫顽力估计。我们推导出与显式正则边界相关的完全非线性模型接近平衡的经典解的存在，并且我们获得了在这种扰动设置下指数收敛到平衡的速率的显式估计。证明基于线性能量方法，该方法将速度空间中碰撞算子的矫顽力性质与输运效应相结合，以推导整个相空间中的矫顽力估计。
For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.