Averaging lemmas with a force term in the transport equation

Abstract

We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We characterize the optimal conditions of these assumptions to compare the obtained regularities according to the space and velocity variables. Our results are mainly in L 2 , and for constant force, in L p for 1 < p ≤ 2. RésuméNous obtenons plusieurs lemmes de moyenne pour deséquations de transport avec un terme de force. Ces résultats améliorent la régularité connue en ne considérant pas le terme de force comme un terme source arbitraire. Deux techniques sont utilisées : des changements de variables locaux ou des phases stationnaires. Ces résultats sont quantifiées par deux hypothèses de non dégénérescence. Nous caractérisons les conditions optimales de ces hypothèses pour comparer les régularités obtenues, par rapport aux variables d'espace et de vitesse. Les résultats sont principalement dans L 2 , et pour le cas constant, dans L p pour 1 < p ≤ 2.

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