Moments and regularity for a Boltzmann equation via Wigner transform
Moments and regularity for a Boltzmann equation via Wigner transform
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DOI：
10.3934/dcds.2019204
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发表时间：
2018-04
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通讯作者：
Thomas Chen;Ryan Denlinger;N. Pavlović
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作者：
Thomas Chen;Ryan Denlinger;N. Pavlović
在本文中，我们继续我们的研究玻尔兹曼方程的工具，源于量子动力学中的色散方程的分析。具体来说，我们专注于碰撞核等于一个常数在空间域$\mathbb{R}^d$，$d\geq 2$，我们在本文中使用的模型的玻尔兹曼方程的解决方案的属性。这个方程的局部适定性已经用维格纳变换证明，当$\left ^\beta f_0 \in L^2_v H^\alpha_x$ for $\min（\alpha，\beta）> \frac{d-1}{2}$时。我们证明了如果$\alpha，\beta$足够大，则可以在$x$中传播矩，在$v$中传播导数（例如，如果$f_0$足够好，则$\left ^k \left ^\ell f \in L^\infty_T L^2_{x，v}$）。该机制是交换的规则性，以换取矩的（逆）维格纳变换的$f$。我们还证明了Sobolev空间$H^{\alpha，\beta}$的正则性结果的持久性;以及解映射在$H^{\alpha，\beta}$中的连续性。总而言之，这些结果使我们能够得出结论的非负性的解决方案，能量守恒，和$H$-定理充分经常的解决方案，通过维格纳变换。特别是非负性被证明对任何$\alpha，\beta > \frac{d-1}{2}$在$H^{\alpha，\beta}$中成立，没有任何额外的正则性或衰减假设。
In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $\mathbb{R}^d$, $d\geq 2$, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $\left ^\beta f_0 \in L^2_v H^\alpha_x$ for $\min (\alpha,\beta) > \frac{d-1}{2}$. We prove that if $\alpha,\beta$ are large enough, then it is possible to propagate moments in $x$ and derivatives in $v$ (for instance, $\left ^k \left ^\ell f \in L^\infty_T L^2_{x,v}$ if $f_0$ is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of $f$. We also prove a persistence of regularity result for the scale of Sobolev spaces $H^{\alpha,\beta}$; and, continuity of the solution map in $H^{\alpha,\beta}$. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the $H$-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in $H^{\alpha,\beta}$ for any $\alpha,\beta > \frac{d-1}{2}$, without any additional regularity or decay assumptions.