Local Well-Posedness for the Boltzmann Equation with Very Soft Potential and Polynomially Decaying Initial Data
Local Well-Posedness for the Boltzmann Equation with Very Soft Potential and Polynomially Decaying Initial Data
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DOI：
10.1137/21m1427504
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发表时间：
2021-06
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通讯作者：
Christopher Henderson;W. Wang
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文献类型：
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作者：
Christopher Henderson;W. Wang
本文讨论了空间非齐次非截断Boltzmann方程在初始数据以速度为变量多项式衰减时的局部适定性。我们考虑的情况下，非常软的潜力$\gamma + 2s-3/2$以前的作品。我们的方法是完全基于碰撞算子的Carleman分解成一个较低的阶项和积分微分算子类似的分数拉普拉斯算子。有趣的是，当$\gamma \in（-3，0]$和$s \in（0，1/2）$在一个我们也包含的加权C^1 $空间中时，这给出了一个非常简短的局部适定性证明。
In this paper, we address the local well-posedness of the spatially inhomogeneous non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $\gamma + 2s-3/2$ of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when $\gamma \in (-3,0]$ and $s \in (0,1/2)$ in a weighted $C^1$ space that we include as well.