Recent progress in velocity averaging

Arsénio, Diogo

Journées équations aux dérivées partielles (2015), article no. 1, 17 p.

A classical result in kinetic theory establishes that if $f(x,v)$ and $v·{\nabla }_{x}f(x,v)$ both belong to $L^2\left({ℝ}_x^n\times {ℝ}_v^n\right)$, then ${\int }_{K}fdv\in H^{\frac{1}{2}}\left({ℝ}_x^n\right)$, for any compact set $K\subset {ℝ}_v^n$. Such regularity statements are known as velocity averaging lemmas and have important implications in the analysis of kinetic equations.

It was asked in [2] whether other settings of velocity averaging could produce a similar maximal gain of regularity of half a derivative. This question, motivated by an earlier work of Pierre-Emmanuel Jabin and Luis Vega [17] on the subject, turns out to be surprisingly rich and difficult, and it is, for the moment, far from being fully understood.

In this article, after recalling some classical results in the field, we survey the recent developments from [2], where new settings of velocity averaging lemmas were investigated. We also formulate a few conjectures, mainly derived from a dimensional analysis and by analogy with known results, thus delimiting the possibilities for other new settings of velocity averaging.

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