New Results in Velocity Averaging

Golse, François

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2001-2002), Exposé no. 9, 15 p.

This paper discusses two new directions in velocity averaging. One is an improvement of the known velocity averaging results for $L^1$ functions. The other shows how to adapt some of the ideas of velocity averaging to a situation that is essentially a new formulation of the Vlasov-Maxwell system.

[1] C. Bardos, P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 101–118. | Numdam | MR | Zbl

[2] C. Bardos, F. Golse, C.D. Levermore, Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation, Commun. Pure & Appl. Math 46 (1993), 667–753. | MR | Zbl

[3] F. Bouchut, L. Desvillettes, Averaging lemmas without time Fourier transform and application to discretized kinetic equations, Proc. Royal Soc. Edinburgh 129A (1999), 19–36. | MR | Zbl

[4] F. Bouchut, F. Golse, C. Pallard, Nonresonant smoothing for wave+transport systems and the Vlasov-Maxwell system, to appear in the proceedings of the IMA, Springer-Verlag. | MR

[5] F. Bouchut, F. Golse, C. Pallard, Nonresonant velocity averaging for wave+transport systems, in preparation.

[6] F. Bouchut, F. Golse, C. Pallard, Conditional regularity of solutions to the 3D Vlasov-Maxwell system, in preparation.

[7] F. Bouchut, F. Golse, M. Pulvirenti, Kinetic Equations and Asymptotic Theory, B. Perthame and L. Desvillettes eds., Series in Applied Mathematics 4, Gauthier-Villars, Paris, 2000. | MR | Zbl

[8] F. Castella, B. Perthame, Estimations de Strichartz pour les équations de transport, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 535–540. | MR | Zbl

[9] R. DeVore, G. Petrova, The averaging lemma, J. Amer. Math. Soc. 14 (2001), 279–296. | MR | Zbl

[10] R. DiPerna, P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), no. 6, 729–757. | MR | Zbl

[11] R.J. DiPerna and P.-L. Lions, On the Cauchy Problem for the Boltzmann Equation: Global Existence and Weak Stability Results, Annals of Math. 130 (1990), 321–366. | MR | Zbl

[12] R. DiPerna, P.-L. Lions, Y. Meyer, $L^p$ regularity of velocity averages Ann. Inst. H. Poincaré Anal. Non Lin. 8 (1991), 271–288. | Numdam | MR | Zbl

[13] N. Dunford, J. T. Schwartz Linear operators, part I, Interscience Publishers Inc., New York 1958. | MR | Zbl

[14] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations 16, (1991), 1761–1794. | MR | Zbl

[15] P. Gérard, F. Golse, Averaging regularity results for PDEs under transversality assumptions Comm. Pure Appl. Math. 45 (1992), 1–26. | MR | Zbl

[16] R. Glassey, W. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal. 92 (1986), 59–90. | MR | Zbl

[17] R. Glassey, W. Strauss, High Velocity Particles in a Collisionless Plasma, Math. Meth. Appl. Sci. 9 (1987), 46–52. | MR | Zbl

[18] R. Glassey, W. Strauss, Absence of Schocks in an Initially Dilute Collisionless Plasma, Comm. Math. Phys. 113 (1987), 191–208. | MR | Zbl

[19] F. Golse Quelques résultats de moyennisation pour les équations aux dérivées partielles in Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino 1988, Special Issue, 101–123 (1989). | MR | Zbl

[20] 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), 110–125. | MR | Zbl

[21] F. Golse, B. Perthame, R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale de l’opérateur de transport, C.R. Acad. Sci. Paris série I 301 (1985), 341–344. | Zbl

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

[23] F. Golse, L. Saint-Raymond, The Navier-Stokes limit for the Boltzmann equation: convergence proof, preprint; & C.R. Acad. Sci. Paris série I 333 (2001), 897–902. | MR | Zbl

[24] D. Hilbert, Begründung der kinetischen Gastheorie, Math. Annalen 72 (1912), 562–578. | MR

[25] S. Klainerman, G. Staffilani, A new approach to the Maxwell Vlasov equations, preprint.

[26] L. Landau, E. Lifshitz, Cours de physique théorique. Vol. 2: Théorie des champs, Editions Mir, Moscou, 1970.

[27] J.-L. Lions, Théorèmes de trace et d’interpolation I, II, Ann. Scuola Norm. di Pisa 13 (1959), pp. 389–403, 14 (1960), pp. 317–331. | Numdam | Zbl

[28] P.-L. Lions Régularité optimale des moyennes en vitesse, C. R. Acad. Sci. Paris Série I 320 (1995), 911–915 & C. R. Acad. Sci. Paris Série I 326 (1998), 945–948. | MR | Zbl

[29] P.-A. Meyer; Probabilités et potentiel, Hermann, Paris 1966. | MR | Zbl

[30] P.-L. Lions, T. Paul Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9, (1993), 553–618. | MR | Zbl

[31] B. Perthame, Global Existence to the BGK Model of the Boltzmann Equation, J. Diff. Eq. 82 (1989), 191–205. | MR | Zbl

[32] B. Perthame, P. Souganidis, A limiting case for velocity averaging, Ann. Scient. Ecole Normale Sup. 4ème série 31, (1998), 591–598. | Numdam | MR | Zbl

[33] K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), 281–303. | MR | Zbl

[34] L. Saint-Raymond, Discrete time Navier-Stokes limit for the BGK Boltzmann equation, Comm. Partial Differential Equations 27 (2002), 149–185. | MR | Zbl

[35] C. Villani, Limites hydrodynamiques de l’équation de Boltzmann [d’après C. Bardos, F. Golse, D. Levermore, P.-L. Lions, N. Masmoudi, L. Saint-Raymond], Séminaire Bourbaki, vol. 2000-2001, Exp. 893. | Numdam