Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article suivant
Diogo Arsénio
Recent progress in velocity averaging
Journées équations aux dérivées partielles (2015), Exp. No. 1, 17 p., doi: 10.5802/jedp.630
Article PDF

Résumé - Abstract

A classical result in kinetic theory establishes that if $f(x,v)$ and $v\cdot \nabla _x f(x,v)$ both belong to $L^2\left(\mathbb{R}^n_x\times \mathbb{R}^n_v\right)$, then $\int _{K}fdv\in H^\frac{1}{2}\left(\mathbb{R}^n_x\right)$, for any compact set $K\subset \mathbb{R}^n_v$. 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.

Bibliographie

[1] V. I. Agoshkov, “Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation”, Dokl. Akad. Nauk SSSR 276 (1984) no. 6, p. 1289-1293  MR 753365 |  Zbl 0599.35009
[2] Diogo Arsénio & Nader Masmoudi, “Maximal gain of regularity in velocity averaging lemmas”, submitted for publication, 2015
[3] Diogo Arsénio & Laure Saint-Raymond, “Compactness in kinetic transport equations and hypoellipticity”, J. Funct. Anal. 261 (2011) no. 10, p. 3044-3098 Article |  MR 2832590 |  Zbl 1231.42023
[4] Jöran Bergh & Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223  MR 482275 |  Zbl 0344.46071
[5] Max Bézard, “Régularité $L^p$ précisée des moyennes dans les équations de transport”, Bull. Soc. Math. France 122 (1994) no. 1, p. 29-76 Numdam |  MR 1259108 |  Zbl 0798.35025
[6] F. Bouchut, “Hypoelliptic regularity in kinetic equations”, J. Math. Pures Appl. (9) 81 (2002) no. 11, p. 1135-1159 Article |  MR 1949176 |  Zbl 1045.35093
[7] François Castella & Benoît Perthame, “Estimations de Strichartz pour les équations de transport cinétique”, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) no. 6, p. 535-540  MR 1383431 |  Zbl 0848.35095
[8] P. Constantin & J.-C. Saut, “Local smoothing properties of dispersive equations”, J. Amer. Math. Soc. 1 (1988) no. 2, p. 413-439 Article |  MR 928265 |  Zbl 0667.35061
[9] Ronald DeVore & Guergana Petrova, “The averaging lemma”, J. Amer. Math. Soc. 14 (2001) no. 2, p. 279-296 Article |  MR 1815213 |  Zbl 1001.35079
[10] R. J. DiPerna, P.-L. Lions & Y. Meyer, “$L^p$ regularity of velocity averages”, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) no. 3-4, p. 271-287 Numdam |  MR 1127927 |  Zbl 0763.35014
[11] Charles Fefferman, “The multiplier problem for the ball”, Ann. of Math. (2) 94 (1971), p. 330-336  MR 296602 |  Zbl 0234.42009
[12] François Golse, Pierre-Louis Lions, Benoît Perthame & Rémi Sentis, “Regularity of the moments of the solution of a transport equation”, J. Funct. Anal. 76 (1988) no. 1, p. 110-125 Article |  MR 923047 |  Zbl 0652.47031
[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. 301 (1985) no. 7, p. 341-344  MR 808622 |  Zbl 0591.45007
[14] Loukas Grafakos, Classical Fourier analysis, Graduate Texts in Mathematics 249, Springer, New York, 2008  MR 2445437 |  Zbl 1220.42001
[15] Loukas Grafakos, Modern Fourier analysis, Graduate Texts in Mathematics 250, Springer, New York, 2009 Article |  MR 2463316 |  Zbl 1158.42001
[16] Pierre-Emmanuel Jabin & Luis Vega, “Averaging lemmas and the X-ray transform”, C. R. Math. Acad. Sci. Paris 337 (2003) no. 8, p. 505-510 Article |  MR 2017127 |  Zbl 1030.35005
[17] Pierre-Emmanuel Jabin & Luis Vega, “A real space method for averaging lemmas”, J. Math. Pures Appl. (9) 83 (2004) no. 11, p. 1309-1351 Article |  MR 2096303 |  Zbl 1082.35043
[18] Pierre-Louis Lions, “Régularité optimale des moyennes en vitesses”, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) no. 8, p. 911-915 Article |  MR 1328710 |  Zbl 0827.35110
Copyright Cellule MathDoc 2018 | Crédit | Plan du site