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

2017/03/31 by Fournier, Nicolas
#60K35 #82C40 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1703.10874

Abstract

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution (f_t)_{t≥ 0}, once the initial condition f₀ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a (0,∞)×ℝ³ random variable (M_t,V_t) such that E[M_t \bf 1_\V_t ∈ ⋅\]=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 f₀, 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 and of its interpretation by McKean.

Related

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• Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
• Mathematical treatment of the homogeneous Boltzmann equation for Maxwellian molecules in the presence of singular kernels
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