Abstract
We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.
Highlights
2010 Mathematics Subject Classification: 35D10 (Primary); 35B65, 35Q20, 82B40 (Secondary) Keywords: Smoothing of weak solutions, Non-cutoff homogeneous Boltzmann equation, DebyeYukawa potential, Maxwellian molecules
Let f be a weak solution of the Cauchy problem (1) for the homogeneous Bolzmann equation for Maxwellian molecules with angular collision kernel satisfying (6) and (8), and initial datum f0 ≥ 0, f0 ∈ L12(Rd) ∩ L log L(Rd)
We follow the strategy we developed in [4], where an inductive procedure was invented to control the commutation error, in order to prove the Gevrey smoothing conjecture in the Maxwellian molecules case
Summary
H is increasing, concave and for any 0 ≤ s− ≤ s+, μ+1 h(s− + s+) ≤ 1 + log α h(s−) + h(s+). S− ≤ s+ Lemma 2.1 shows that the subadditivity bound can be improved to gain the small factor μ+1 1+log α. Since h′(s) = μ + 1 log(α + s) μ ≥ 0 α+s the function h is increasing. 1 d+2 can be improved if higher derivatives of the function h are bounded, see Section 2.3 in [4] This was important for the results of [4], but we don’t need it here because of the stronger form of the enhanced subadditivity Lemma for the weight we consider in this paper.
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