A new commutator method for averaging lemmas

Jabin, Pierre-Emmanuel; Lin, Hsin-Yi; Tadmor, Eitan

doi:10.5802/slsedp.142

A new commutator method for averaging lemmas

Jabin, Pierre-Emmanuel ¹ ; Lin, Hsin-Yi ² ; Tadmor, Eitan ³

¹ Pennsylvania State University, Department of Mathematics and Huck Institutes, State College, PA 16802, USA
² CIRES, University of Colorado Boulder, CO 80309, USA
³ Department of Mathematics and Institute for Physical Sciences & Technology (IPST), University of Maryland, College Park, MD 20742, USA

Séminaire Laurent Schwartz — EDP et applications (2019-2020), Exposé no. 10, 19 p.

Résumé

This document corresponds to the talk that the first author gave at the Laurent Schwartz seminar on March 10th 2020. It introduces, in a simplified setting, a novel commutator method to obtain averaging lemma estimates. Averaging lemmas are a type regularizing effect on averages in velocity of solutions to kinetic equations. We introduce a new bilinear approach that naturally leads to velocity averages in $L^{2} ([0, T], H_{x}^{s})$ . The new method outperforms classical averaging lemma results when the right-hand side of the kinetic equation has enough integrability. It also allows a perturbative approach to averaging lemmas which provides, for the first time, explicit regularity results for non-homogeneous velocity fluxes.

Publié le : 2022-02-04

DOI : 10.5802/slsedp.142

Affiliations des auteurs :

Jabin, Pierre-Emmanuel ¹ ; Lin, Hsin-Yi ² ; Tadmor, Eitan ³

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Jabin, Pierre-Emmanuel; Lin, Hsin-Yi; Tadmor, Eitan. A new commutator method for averaging lemmas. Séminaire Laurent Schwartz — EDP et applications (2019-2020), Exposé no. 10, 19 p. doi : 10.5802/slsedp.142. http://www.numdam.org/articles/10.5802/slsedp.142/

Bibliographie
Cité par

[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(6):1289-1293, 1984.

[2] D. Arsénio, N. Lerner. An energy method for averaging lemmas. | arXiv

[3] D. Arsénio, L. Saint-Raymond. Compactness in kinetic transport equations and hypoellipticity. J. Funct. Anal., 261(10), (2011), 3044-3098. | DOI | MR | Zbl

[4] D. Arsénio, N. Masmoudi, A new approach to velocity averaging lemmas in Besov spaces, J. Math. Pures Appl.(9) 101 (2014), no. 4, 495-551. | DOI | MR | Zbl

[5] D. Arsénio, N. Masmoudi. Maximal gain of regularity in velocity averaging lemmas, Analysis and PDE, Vol. 12 (2019), No. 2, 333-388. | DOI | MR | Zbl

[6] M. Bézard, Régularité $L^{p}$ précisée des moyennes dans les équations de transport, Bull. Soc. Math. France, 122(1) (1994) 29-76. | DOI | Zbl

[7] F. Bouchut, Hypoelliptic regularity in kinetic equations, J. Math. Pures Appl. (9) 81 (11) (2002) 1135–1159. | DOI | MR | Zbl

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Viriel, Morawetz, and interaction Morawetz inequalities.

[9] R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511-547. | DOI | MR | Zbl

[10] R.J. DiPerna, P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729-757. | DOI | MR | Zbl

[11] R. J. DiPerna, P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math. .2/ 130:2 (1989), 321-366. | DOI | MR | Zbl

[12] 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, 271-287. | DOI | Zbl

[13] C. De Lellis, M. Westdickenberg, On the optimality of velocity averaging lemmas, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 6, 1075-1085. | DOI | MR | Zbl

[14] R. DeVore, G. Petrova, The averaging lemma, J. Amer. Math. Soc. 14 (2001), no. 2, 279-296. | DOI | MR | Zbl

[15] S. Doi, On the Cauchy problem for Schrödinger type equations and the regularity of the solutions. J. Math. Kyoto Univ. 34 (1994), no. 2, 319-328. | DOI | Zbl

[16] D. Han-Kwan, $L^{1}$ averaging lemma for transport equations with Lipschitz force fields, Kinet. Relat. Models 3 (4) (2010) 669-683. | DOI | MR | Zbl

[17] M. Escobedo, S. Mischler and M. Valle (2003), Homogeneous Boltzmann equation in quantum relativistic kinetic theory, Vol. 4 of Electronic Journal of Differential Equations. Monograph, Southwest Texas State University, San Marcos, TX.

[18] F. Golse, P.-L. Lions, B. Perthame, R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110-125. | DOI | MR | Zbl

[19] F. Golse, F. Poupaud, Limite fluide des équations de Boltzmann des semiconducteurs pour une statistique de Fermi-Dirac, Asymptotic Anal. 6 (1992) 135-160. | DOI | Zbl

[20] F. Golse, L. Saint-Raymond, Hydrodynamic limits for the Boltzmann equation, Riv. Mat. Univ. Parma 4 (2005), 1-144. | Zbl

[21] F. Golse, B. Perthame, Optimal regularizing effect for scalar conservation laws, Rev. Mat. Iberoam., 29 (2013), no. 4, 1477-1504. | DOI | MR | Zbl

[22] F. Golse and L. Saint-Raymond, Velocity averaging in $L^{1}$ for the transport equation, C. R. Acad. Sci. Paris Ser. I Math, 334 (2002), 557-562. | DOI | MR | Zbl

[23] F. Golse, Fluid dynamic limits of the kinetic theory of gases, From particle systems to partial differential equations. Springer, Berlin, Heidelberg, (2014), 3-91. | DOI | Zbl

[24] I. Gasser, P. Markowich, B. Perthame, Dispersion and Moments Lemma revisited, J. Differential Equations 156 (1999), 254-281. | DOI | MR | Zbl

[25] L. Hörmander, Hypoelliptic second order differential equations. Acta Mathematica 119.1 (1967) 147-171. | DOI | MR | Zbl

[26] P.-E. Jabin, H.-Y. Lin, E. Tadmor, Commutator Method for Averaging Lemmas. | arXiv

[27] P.-E. Jabin, H.-Y. Lin, E. Tadmor, Averaging Lemmas with inhomogeneous velocity fluxes. In preparation.

[28] P.-E. Jabin, B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging, Comm. Pure Appl. Math. 54 (9) (2001) 1096- 1109. | DOI | MR | Zbl

[29] K. Kunihiko, The Cauchy problem for Schrödinger type equations with variable coefficients. J. Math. Soc. Japan 50 (1998), no. 1, 179-202. | DOI | Zbl

[30] P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7 (1994), no. 1, 169-191. | DOI | MR | Zbl

[31] P.-L. Lions, B. Perthame, E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems, Comm. Math. Phys. 163 (1994) 415-431 | DOI | Zbl

[32] N. Masmoudi, M. L. Tayeb, Diffusion limit of a semiconductor Boltzmann-Poisson system. SIAM Journal on Mathematical Analysis 38.6 (2007) 1788-1807. | DOI | MR | Zbl

[33] B. Perthame, Kinetic formulation of conservation laws. Oxford Lecture Series in Mathematics and Its Applications, 21. Oxford University Press, Oxford, (2002). | Zbl

[34] G. Staffilani, D. Tataru, Strichartz estimates for a Schrödinger operator with nonsmooth coefficients, Comm. Partial Differential Equations 27 (2002), no. 7-8, 1337-1372 | DOI | Zbl

[35] E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., (1970). | Zbl

[36] L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Springer Science and Business Media, no. 1971, (2009).

[37] E. Tadmor, T. Tao, Velocity averaging, kinetic formulations, and regularizing effects in quasi-linear PDEs, Comm. Pure Appl. Math. 60 (2007), no. 10, 1488-1521. | DOI | MR | Zbl

[38] M. Westdickenberg, Some new velocity averaging results. SIAM J. Math. Anal. 33 (2002), no. 5, 1007-1032. | DOI | MR | Zbl

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