Local well-posedness of the Boltzmann equation with polynomially decaying initial data
Local well-posedness of the Boltzmann equation with polynomially decaying initial data
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DOI：
10.3934/krm.2020029
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发表时间：
2019-10
影响因子：
1
通讯作者：
Christopher Henderson;Stanley Snelson;Andrei Tarfulea
中科院分区：
文献类型：
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作者：
Christopher Henderson;Stanley Snelson;Andrei Tarfulea
我们考虑空间非齐次非截止玻尔兹曼方程的柯西问题，其中速度变量的初始数据呈多项式衰减。我们通过在混合 $L^2$ 和 $L^\infty$ 空间中工作，在五阶 Sobolev 空间中建立具有这种衰减的任何初始数据的短时存在性，该空间允许补偿潜在的矩生成并获得非常适合该空间的碰撞算子的新估计。我们的结果改进了玻尔兹曼方程在这种衰变机制中适定的参数范围，以及对初始正则性的限制。作为一种应用，我们可以将我们的存在性结果与最近的 Imbert-Silvestre 条件正则性估计 (arXiv:1909.12729 [math.AP]) 结合起来，得出只要质量、能量和熵密度保持在控制范围内，解就可以继续存在的结论。此延续标准以前仅适用于 $0 的情况
We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed $L^2$ and $L^\infty$ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities stay under control. This continuation criterion was previously only available in the case $0