×

Velocity averaging in \(L^1\) for the transport equation. (English. Abridged French version) Zbl 1154.35326

Summary: A new result of \(L^1\)-compactness for velocity averages of solutions to the transport equation is stated and proved in this Note. This result, proved by a new interpolation argument, extends to the case of any space dimension Lemma 8 of F. Golse, P. L. Lions, B. Perthame and R. Sentis, J. Funct. Anal. 76, 110–125 (1988; Zbl 0652.47031)], proved there in space dimension 1 only. This is a key argument in the proof of the hydrodynamic limits of the Boltzmann or BGK equations to the incompressible Euler or Navier-Stokes equations.

MSC:

35F20 Nonlinear first-order PDEs
35Q35 PDEs in connection with fluid mechanics
82C70 Transport processes in time-dependent statistical mechanics

Citations:

Zbl 0652.47031
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agoshkov, V.I., Spaces of functions with differential-difference characteristics and smoothness of solutions of the transport equation, Soviet math. dokl., 29, 662-666, (1984) · Zbl 0599.35009
[2] Bardos, C.; Degond, P., Global existence for the vlasov – poisson equation in 3 space variables with small initial data, Ann. inst. H. Poincaré anal. non linéaire, 2, 101-118, (1985) · Zbl 0593.35076
[3] Bouchut, F.; Golse, F.; Pulvirenti, M., Kinetic equations and asymptotic theory, ()
[4] Castella, F.; Perthame, B., Estimations de Strichartz pour LES équations de transport cinétique, C. R. acad. sci. Paris, Série I, 322, 535-540, (1996) · Zbl 0848.35095
[5] DeVore, R.; Petrova, G., The averaging lemma, J. amer. math. soc., 14, 279-296, (2001) · Zbl 1001.35079
[6] DiPerna, R.; Lions, P.-L.; Meyer, Y., L^{p} regularity of velocity averages, Ann. inst. H. Poincaré anal. non linéaire, 8, 271-288, (1991) · Zbl 0763.35014
[7] Golse, F.; Lions, P.-L.; Perthame, B.; Sentis, R., Regularity of the moments of the solution of a transport equation, J. funct. anal., 76, 110-125, (1988) · Zbl 0652.47031
[8] Golse, F.; Perthame, B.; Sentis, R., 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, 341-344, (1985) · Zbl 0591.45007
[9] 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), to appear · Zbl 1056.35134
[10] F. Golse, L. Saint-Raymond, in preparation
[11] Lions, J.-L., Théorèmes de trace et d’interpolation I, II, Ann. scuola norm. Pisa, 13, 389-403, (1959), 14 (1960) 317-331 · Zbl 0097.09502
[12] Lions, P.-L., Régularité optimale des moyennes en vitesse, C. R. acad. sci. Paris, Série I, 320, 911-915, (1995), and C. R. Acad. Sci. Paris, Série I 326 (1998) 945-948 · Zbl 0827.35110
[13] Meyer, P.-A., Probabilités et potentiel, (1966), Hermann Paris
[14] L. Saint-Raymond, Thèse de doctorat en mathématiques, Université Paris VII-Denis Diderot, January 2000
[15] L. Saint-Raymond, From the Boltzmann BGK equation to the Navier-Stokes system, Ann. Sci. École Norm. Sup., in press · Zbl 1009.35071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.