AI 中文总结
针对截断麦克斯韦分子的三维空间非均匀Boltzmann方程，构造了具有奇异无限质量初始数据的全局前向自相似温和解，通过双线性估计和Kaniel--Shinbrot迭代证明其存在性。
AI 中文摘要
对于每个固定的$m>1$，我们构造了三维空间非均匀麦克斯韦分子Boltzmann方程的非零全局前向自相似温和解。我们假设角核有界，并满足在掠射角和迎面角处的加权可积性条件。初始数据在动力学标度下是齐次的，在相空间原点处奇异，且具有无限质量。它们的大小在一个在此标度下不变的加权范数中是小量。我们首先利用自由输运坐标和碰撞位移的正交性，证明了该范数下增益算子的双线性估计。此估计给出了无损失项方程的全局解，该解界定了Kaniel--Shinbrot迭代中的上下近似。我们证明在初始时刻，在相空间的一个固定环带上，它们的差以任意高阶趋于零。论证使用了有限次碰撞估计的迭代，并在特征线离开空间环带时停止特征线。自相似性随后给出该差的有限质量和有限的一阶绝对速度矩。将其质量平衡与标度律比较表明两个极限一致。所得解在相空间远离原点处局部地在$L^\infty$中达到其初始迹，并在每个正时刻具有无限质量。
英文摘要
For every fixed $m>1$, we construct nonzero global forward self-similar mild solutions of the three-dimensional spatially inhomogeneous Boltzmann equation for Maxwell molecules. We assume that the angular kernel is bounded and satisfies a weighted integrability condition at the grazing and head-on angles. The initial data are homogeneous under the kinetic scaling, singular at the origin in phase space, and have infinite mass. Their size is small in a weighted norm invariant under this scaling. We first prove a bilinear estimate for the gain operator in this norm, using free-transport coordinates and the orthogonality of the collision displacements. This estimate gives a global solution of the equation without the loss term, which bounds the lower and upper approximations in the Kaniel--Shinbrot iteration. We prove that their difference tendsto zero to arbitrarily high order at the initial time on a fixed annulus in phase space. The argument uses a finite number of iterations of the collision estimate and stops characteristics when they leave a spatial annulus. Self-similarity then gives finite mass and a finite first absolute velocity moment for the difference. Comparing its mass balance with its scaling law shows that the two limits agree. The resulting solution attains its initial trace locally in $L^\infty$ away from the origin in phase space and has infinite mass at every positive time.
