Recent progress in velocity averaging
Diogo Arsénio1
1 Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR CNRS 7586) Université Paris Diderot (Paris 7) 75013 Paris France
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·∇ x f(x,v) both belong to L 2 ℝ x n ×ℝ v n , then ∫ K fdv∈H 1 2 ℝ x n , for any compact set K⊂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.

DOI : 10.5802/jedp.630
Diogo Arsénio 1

1 Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR CNRS 7586) Université Paris Diderot (Paris 7) 75013 Paris France 
@incollection{JEDP_2015____A1_0,
     author = {Diogo Ars\'enio},
     title = {Recent progress in velocity averaging},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {1},
     pages = {1--17},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2015},
     doi = {10.5802/jedp.630},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.630/}
}
TY  - JOUR
AU  - Diogo Arsénio
TI  - Recent progress in velocity averaging
JO  - Journées équations aux dérivées partielles
PY  - 2015
SP  - 1
EP  - 17
PB  - Groupement de recherche 2434 du CNRS
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.630/
DO  - 10.5802/jedp.630
LA  - en
ID  - JEDP_2015____A1_0
ER  - 
%0 Journal Article
%A Diogo Arsénio
%T Recent progress in velocity averaging
%J Journées équations aux dérivées partielles
%D 2015
%P 1-17
%I Groupement de recherche 2434 du CNRS
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.630/
%R 10.5802/jedp.630
%G en
%F JEDP_2015____A1_0
Diogo Arsénio. Recent progress in velocity averaging. Journées équations aux dérivées partielles (2015), article  no. 1, 17 p. doi : 10.5802/jedp.630. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.630/

[1] V. I. Agoshkov Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR, Volume 276 (1984) no. 6, pp. 1289-1293 | MR | Zbl

[2] Diogo Arsénio; Nader Masmoudi Maximal gain of regularity in velocity averaging lemmas (2015) (submitted for publication)

[3] Diogo Arsénio; Laure Saint-Raymond Compactness in kinetic transport equations and hypoellipticity, J. Funct. Anal., Volume 261 (2011) no. 10, pp. 3044-3098 | DOI | MR | Zbl

[4] Jöran Bergh; Jörgen Löfström Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, pp. x+207 (Grundlehren der Mathematischen Wissenschaften, No. 223) | MR | Zbl

[5] Max Bézard Régularité L p précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, Volume 122 (1994) no. 1, pp. 29-76 | Numdam | MR | Zbl

[6] F. Bouchut Hypoelliptic regularity in kinetic equations, J. Math. Pures Appl. (9), Volume 81 (2002) no. 11, pp. 1135-1159 | DOI | MR | Zbl

[7] François Castella; Benoît Perthame Estimations de Strichartz pour les équations de transport cinétique, C. R. Acad. Sci. Paris Sér. I Math., Volume 322 (1996) no. 6, pp. 535-540 | MR | Zbl

[8] P. Constantin; J.-C. Saut Local smoothing properties of dispersive equations, J. Amer. Math. Soc., Volume 1 (1988) no. 2, pp. 413-439 | DOI | MR | Zbl

[9] Ronald DeVore; Guergana Petrova The averaging lemma, J. Amer. Math. Soc., Volume 14 (2001) no. 2, pp. 279-296 | DOI | MR | Zbl

[10] R. J. DiPerna; P.-L. Lions; Y. Meyer L p regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 8 (1991) no. 3-4, pp. 271-287 | Numdam | MR | Zbl

[11] Charles Fefferman The multiplier problem for the ball, Ann. of Math. (2), Volume 94 (1971), pp. 330-336 | MR | Zbl

[12] François Golse; Pierre-Louis Lions; Benoît Perthame; Rémi Sentis Regularity of the moments of the solution of a transport equation, J. Funct. Anal., Volume 76 (1988) no. 1, pp. 110-125 | DOI | MR | Zbl

[13] François Golse; Benoît Perthame; Rémi Sentis Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math., Volume 301 (1985) no. 7, pp. 341-344 | MR | Zbl

[14] Loukas Grafakos Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, New York, 2008, pp. xvi+489 | MR | Zbl

[15] Loukas Grafakos Modern Fourier analysis, Graduate Texts in Mathematics, 250, Springer, New York, 2009, pp. xvi+504 | DOI | MR | Zbl

[16] Pierre-Emmanuel Jabin; Luis Vega Averaging lemmas and the X-ray transform, C. R. Math. Acad. Sci. Paris, Volume 337 (2003) no. 8, pp. 505-510 | DOI | MR | Zbl

[17] Pierre-Emmanuel Jabin; Luis Vega A real space method for averaging lemmas, J. Math. Pures Appl. (9), Volume 83 (2004) no. 11, pp. 1309-1351 | DOI | MR | Zbl

[18] Pierre-Louis Lions Régularité optimale des moyennes en vitesses, C. R. Acad. Sci. Paris Sér. I Math., Volume 320 (1995) no. 8, pp. 911-915 | DOI | MR | Zbl

Cité par Sources :