Averaging regularity results for PDEs under transversality assumptions. (English) Zbl 0832.35020

Summary: Let \(u= u(x, y)\) and \(f= f(x, y)\) be two functions related by a PDE \(P(x, y, D_x, D_y) u= f\); the regularity of the \(y\)-average \(\int u(x,\cdot) dy\) as a function of \(x\) is investigated knowing that of \(u\) and \(f\). Our method consists in reducing \(P\) to a microlocal normal form under a natural transversality assumption. The 2-microlocal regularity of \(u\) is also determined knowing that of \(f\). These results are then applied to a homogenization problem. This article generalizes the results of [the second author, P.-L. Lions, B. Perthame and R. Sentis, J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)] and [the second author, C. R. Acad. Sci., Paris, Ser. I 305, 801-804 (1987; Zbl 0658.35083)] on velocity averaging and homogenization for kinetic equations.


35B65 Smoothness and regularity of solutions to PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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[1] Bardos, C. R. Acad. Sci. 309 pp 727– (1989)
[2] Bardos, J. of Stat. Phys. 63 pp 323– (1991)
[3] , and , Fluid dynamic limits of kinetic equations II: Convergence proofs for the Boltzmann equation, preprint, 1990.
[4] Bardos, J. Funct. Anal. 77 pp 434– (1988)
[5] Second Microlocalization and Propagation of Singularities for Semilinear Hyperbolic Equations, Proceedings of the Symposium on Hyperbolic Equations and Related Topics, Katata and Kyoto, 1984.
[6] Cessenat, C. R. Acad. Sci. 299 pp 831– (1984)
[7] DiPerna, Annals of Math. 130 pp 321– (1990)
[8] DiPerna, Comm. Pure Appl. Math. 42 pp 729– (1989)
[9] Gérard, Ann. Scient. de l’Ecole Normale Supérieure 23 pp 89– (1990)
[10] Compacité par compensation et régularité microlocale, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, exposé 6, 1989.
[11] Gérard, Comm. Partial Diff. Equations (1991)
[12] Golse, C. R. Acad. Sci. 305 pp 801– (1988)
[13] Particle transport in nonhomogeneous media, pp. 152–170 in The Mathematics of Fluid and Plasma Dynamics, , and , eds., Lecture Notes in Math. 1460, Springer-Verlag, Berlin, 1991.
[14] Golse, C. R. Acad. Sci. (1991)
[15] Moyennisation des champs de vecteurs et régularité, Journées E. D. P. de St Jean de Monts, 1990.
[16] Golse, J. Funct. Anal. 76 pp 110– (1988)
[17] and , Limite fluide de l’équation de Boltzmann des semi-conducteurs pour une statistique de Fermi-Dirac, submitted to Asymptotic Anal., 1991.
[18] The Analysis of Linear Partial Differential Operators I-IV, Springer-Verlag, Berlin, 1983–1985. · Zbl 1124.05078
[19] Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. · Zbl 0108.09301
[20] Compensated compactness and applications to partial differential equations, pp. 136–211 in Nonlinear Aalysis and Mechanics IV, ed., Research Notes in Math. 39, Pitman, London, 1979.
[21] Tartar, Proc. Roy. Soc. Edinburgh 115 A pp 193– (1990) · Zbl 0774.35008
[22] Golse, C. R. Acad. Sci. 301 pp 341– (1985)
[23] Agoshkov, Sov. Math. Dokl. 29 pp 662– (1984)
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