Regularity of renormalized solutions in the Boltzmann equation with long‐range interactions

Abstract

It is well-established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long-range interactions, any renormalized solution F .t; x; v/ to the Boltzmann equation satisfies locally and for some s > 0 depending on p. We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff.

Gelfand–Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation

“…The occurrence of hypoelliptic properties for kinetic equations was used and pointed out in many recent works, such as the paper by Arsénio & Saint-Raymond [8], as well as Golse's survey [20]. The work by Alexandre, Morimoto, Ukai, Xu & Yang [4] highlighted the importance of regularization effects for Boltzmann equation (see also [3,7,13,16]). It served as a motivation for us to explore more completely the behaviour of solutions of Kac's equation, a somewhat simplified model of Boltzmann equation but still keeping some of the main features of Boltzmann's.…”

A new approach to velocity averaging lemmas in Besov spaces

“…Essentially, we show here how the methods from Section 4 can be used to gain control over the concentrations in kinetic transport equations through velocity averaging. This application relies on a refined regularity estimate for the Boltzmann equation without cutoff established in [3], wherein the reader will also find a complete discussion of this application to the hydrodynamic limit. An alternative method has been developed in [4].…”

“…Indeed, its use is prevented by the particular structure of the Boltzmann collision operator, which behaves as a nonlinear differential operator for collision kernels without cutoff (cf. [1,2,3]). Consequently, the strategy from [16,17,19] would require a generalization of the preceding theorem to kinetic transport equations (5.2) with velocity derivatives in its right-hand side.…”

Solutions of the Vlasov–Maxwell–Boltzmann system with long-range interactions

“…Their analysis consisted first in exploiting the intricate regularizing (or merely compactifying) effects on the velocity variable due to longrange interactions, thanks to a former study by Alexandre, Desvillettes, Villani and Wennberg [1] on the entropy dissipation bound. Alternately, one could also use the results from [4] to deduce a similar velocity smoothness in a more convenient functional setting. Then, they used standard velocity averaging lemmas (see [7], for instance) to obtain some compactness in time and space.…”