SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 1049-1069, January 2021.
We consider an inverse problem for the Boltzmann equation with nonlinear collision operator in dimensions $n\geq 2$. We show that the kinetic collision kernel can be uniquely determined from the incoming-to-outgoing mappings on the boundary of the domain provided that the kernel satisfies a monotonicity condition. Furthermore, a reconstruction formula is also derived. The key methodology is based on the higher-order linearization scheme to reduce a nonlinear equation into simpler linear equations by introducing multiple small parameters into the original equation.

中文翻译：

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非线性Boltzmann方程中碰撞核的重构

SIAM数学分析杂志，第53卷，第1期，第1049-1069页，2021年1月。
我们考虑具有非线性碰撞算子的Boltzmann方程的反问题，维数为$ n \ geq 2 $。我们表明，只要该碰撞满足单调性条件，就可以从域边界上的传入到传出映射唯一确定动力学碰撞内核。此外，还导出了重建公式。关键方法基于高阶线性化方案，通过将多个小参数引入原始方程，从而将非线性方程简化为简单的线性方程。