# American Institute of Mathematical Sciences

June  2019, 12(3): 483-505. doi: 10.3934/krm.2019020

## A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff

 Sorbonne Université, CNRS, LPSM, UMR 8001, F-75005 Paris, France

Received  March 2017 Revised  February 2018 Published  February 2019

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution $(f_t)_{t\geq 0}$, once the initial condition $f_0$ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a $(0,\infty)\times {\mathbb{R}}^3$ random variable $(M_t,V_t)$ such that $\mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t$. We also write down a series expansion of $f_t$. Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express $f_t$ in terms of $f_0$, we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [18] and of its interpretation by McKean [10,11].

Citation: Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic & Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020
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