Diffusive expansion for solutions of the Boltzmann equation in the whole space
Diffusive expansion for solutions of the Boltzmann equation in the whole space
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玻尔兹曼方程解在全空间的扩散展开
DOI：
10.1016/j.jde.2010.07.024
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发表时间：
2011-01
影响因子：
2.4
通讯作者：
Liu, Shuangqian
中科院分区：
文献类型：
--
作者：
Zhao, Huijiang;Liu, Shuangqian
本文关注的是在指定初始数据的整个空间中重新缩放的玻尔兹曼方程解的扩散展开我们的主要目的是证明整个空间 RN 中重新缩放的玻尔兹曼方程 (0.1) 的解 Fϵ(t,x,v) 的扩散展开的全局有效性，对于所有 t⩾0 且初始数据 F0ϵ(x,v) 满足初始展开这里μ(v)=(2π)−N2exp(−|v|22) 是归一化全局麦克斯韦函数。假设初始展开 F0ϵ(x,v) 的系数 fm(0,x,v)(1⩽m⩽n) 的流体分量具有无散度速度场 um0(x) 以及温度场 θm0(x)，如果我们进一步假设 f1(0,x,v) 的速度-温度场 [u10(x),θ10(x)] 在Hs(RN)(s⩾2(N+n+2))，我们可以通过迭代法和能量法唯一地确定扩散展开式(0.3)中的这些系数fm(t,x,v)(1⩽m⩽n)。这些系数fm(t,x,v)(1⩽m⩽n)的流体动力分量分别满足不可压缩条件、布辛涅斯克关系和/或纳维-斯托克斯-傅里叶系统，而这些系数的微观分量由递归公式确定。与Y.Guo(2006)[18]研究的周期盒内的相应问题相比，这里的主要困难是由于Poincaré不等式在整个空间RN中无效，并且通过使用Riesz势的Lp-Lq估计克服了这个困难。此外，通过利用能量方法，我们还可以推导出这些系数 fm(t,x,v)(1⩽m⩽n) 的某些时空能量估计。一旦扩散展开式 (0.3) 中的系数 fm(t,x,v)(1⩽m⩽n) 被唯一确定并获得了一些精细的估计，则通过统一的非线性能量方法建立关于余数 fnϵ(t,x,v) 的 ϵ 的统一估计，并且这样的估计保证了扩散展开式 (0.3) 在大范围内的有效性，前提是注意对于 m⩾2， um(t,x) 不再是无散向量，值得指出的是，对于 m⩾3，在通过 Riesz 势的 Lp–Lq 估计来推导 pm(t,x) 的某些估计时，我们需要要求 N>2n+2。
This paper is concerned with the diffusive expansion for solutions of the rescaled Boltzmann equation in the whole space with prescribed initial data Our main purpose is to justify the global validity of the diffusive expansion for a solution Fϵ(t,x,v) of the rescaled Boltzmann equation (0.1) in the whole space RNfor all t⩾0 with initial data F0ϵ(x,v) satisfying the initial expansion Here μ(v)=(2π)−N2exp(−|v|22) is a normalized global Maxwellian. Under the assumption that the fluid components of the coefficients fm(0,x,v)(1⩽m⩽n) of the initial expansion F0ϵ(x,v) have divergence-free velocity fields um0(x) as well as temperature fields θm0(x), if we assume further that the velocity-temperature fields [u10(x),θ10(x)] of f1(0,x,v) have small amplitude in Hs(RN)(s⩾2(N+n+2)), we can determine these coefficients fm(t,x,v)(1⩽m⩽n) in the diffusive expansion (0.3) uniquely by an iteration method and energy method. The hydrodynamic component of these coefficients fm(t,x,v)(1⩽m⩽n) satisfies the incompressible condition, the Boussinesq relations and/or the Navier–Stokes–Fourier system respectively, while the microscopic component of these coefficients is determined by a recursive formula. Compared with the corresponding problem inside a periodic box studied in Y. Guo (2006) [18], the main difficulty here is due to the fact that Poincaré's inequality is not valid in the whole space RNand this difficulty is overcome by using the Lp–Lq-estimate on the Riesz potential. Moreover, by exploiting the energy method, we can also deduce certain the space–time energy estimates on these coefficients fm(t,x,v)(1⩽m⩽n). Once the coefficients fm(t,x,v)(1⩽m⩽n) in the diffusive expansion (0.3) are uniquely determined and some delicate estimates have been obtained, the uniform estimates with respect to ϵ on the remainders fnϵ(t,x,v) are then established via a unified nonlinear energy method and such an estimate guarantees the validity of the diffusive expansion (0.3) in the large provided that Notice that for m⩾2, um(t,x) is no longer a divergence-free vector and it is worth to pointing out that, for m⩾3, it was in deducing certain estimates on pm(t,x) by the Lp–Lq-estimate on the Riesz potential that we need to require that N>2n+2.
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影响因子：
3
作者：
Yan Guo
通讯作者：
Yan Guo
影响因子：
3.1
作者：
F. Golse;L. Saint-Raymond
通讯作者：
F. Golse;L. Saint-Raymond
影响因子：
2.5
作者：
P. Lions;N. Masmoudi
通讯作者：
P. Lions;N. Masmoudi
影响因子：
3
作者：
R. Caflisch
通讯作者：
R. Caflisch
DOI：
10.1007/978-1-4899-6381-9
发表时间：
1969
期刊：
--
影响因子：
--
作者：
M. Kogan
通讯作者：
M. Kogan