论文标题
重新审视的毕业生型衰减估计值
Grad-Caflisch pointwise decay estimates revisited
论文作者
论文摘要
In the influential paper \cite{Caflish-1980-CPAM} which was the starting point of the employment of Hilbert expansion method to the rigorous justifications of the fluid limits of the Boltzmann equation, Caflisch discovered an elegant and crucial estimate on each expansion term (Proposition 3.1 in \cite{Caflish-1980-CPAM}).证明基本上依赖于grad \ cite {Grad-1963}的估计，该估计是在$ \ Mathcal {l}^{ - 1} $的点衰减属性上γ\ leq 1 $。 Caflisch的论点需要GRAD估算的指数版本。但是，Grad的原始论文仅在多项式衰减上。在本文中，我们重新访问并提供了Caflisch-Grad类型衰减估计值以及Boltzmann Equaiton的可压缩欧拉限制中的相应应用。主要的新颖性是，对于情况，碰撞内核功率$ - \ frac {3} {2} {2} <γ\ leq 1 $，点式估计值的证明不使用任何衍生物。因此，此估计的潜在应用可能比希尔伯特的扩张更广泛。为了使结果的完整性，我们还使用CASE $ -3 <γ\ leq - \ frac {3} {2} $的衍生物证明了几乎无处不在的估算值。此外，在用于流体限制的应用中，$ \ Mathcal {l}^{ - 1} $和衍生物相对于参数（例如，$（t，x）$，这必须发生在$ \ nathcal {l} $的情况下，在围绕$（t，x）$的local Maxwellian围绕local Maxwellian进行线性化时，这必须发生。我们详细介绍了换向器的估计，这在先前的玻尔兹曼方程流体限制文献中缺少。在Boltzmann方程中所有可压缩的流体限制中都需要此估计值。
In the influential paper \cite{Caflish-1980-CPAM} which was the starting point of the employment of Hilbert expansion method to the rigorous justifications of the fluid limits of the Boltzmann equation, Caflisch discovered an elegant and crucial estimate on each expansion term (Proposition 3.1 in \cite{Caflish-1980-CPAM}). The proof essentially relied on an estimate of Grad \cite{Grad-1963}, which was on the pointwise decay properties of $\mathcal{L}^{-1}$, the pseudo-inverse operator of the linearized Boltzmann collision operator $\mathcal{L}$, for the hard potential collision kernel, i.e. the power $0\leq γ\leq 1$. Caflisch's arguments need the exponential version of Grad's estimate. However, Grad's original paper was only on the polynomial decay. In this paper, we revisit and provide a full proof of the Caflisch-Grad type decay estimates and the corresponding applications in the compressible Euler limit of the Boltzmann equaiton. The main novelty is that for the case collision kernel power $-\frac{3}{2}<γ\leq 1$, the proof of the pointwise estimate does not use any derivatives. So the potential applications of this estimate could be wider than in the Hilbert expansion. For the completeness of the result, we also prove the almost everywhere pointwise estimate using derivatives for the case $-3<γ\leq -\frac{3}{2}$. Furthermore, in the application to fluid limits, $\mathcal{L}^{-1}$ and the derivatives with respect to the parameters (for example, $(t,x)$, this must happen when $\mathcal{L}$ is linearized around local Maxwellian which depends on $(t,x)$) are not commutative. We detailed analyze the estimate of commutators, which was missing in previous literatures of fluid limits of the Boltzmann equation. This estimate is needed in all compressible fluid limits from Boltzmann equation.
