×

zbMATH — the first resource for mathematics

A limiting case for velocity averaging. (English) Zbl 0956.45010
Summary: We complete the theory of velocity averaging lemmas for transport equations by studying the limiting case of a full space derivative in the source term. Although the compactness of averages does not hold any longer, a specific estimate remains which shows compactness of averages in more general situations than those previously known. Our method is based on Calderón-Zygmund theory.

MSC:
45K05 Integro-partial differential equations
82C70 Transport processes in time-dependent statistical mechanics
45M05 Asymptotics of solutions to integral equations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] V. I. AGOSHKOV , Spaces of functions with differential-difference characteristics and smoothness of solutions of the transport equation (Soviet Math. Dokl., Vol. 29, 1984 , pp. 662-666). MR 86a:46033 | Zbl 0599.35009 · Zbl 0599.35009
[2] C. BARDOS , F. GOLSE , B. PERTHAME and R. SENTIS , The nonaccretive radiative transfer equations. Existence of solutions and Rosseland approximation (J. Funct. Anal., Vol. 77, 1988 , pp. 434-460). MR 89f:35174 | Zbl 0655.35075 · Zbl 0655.35075
[3] M. BÉZARD , Régularité précisée des moyennes dans les équations de transport (Bull. Soc. Math. France, Vol. 122, 1994 , pp. 29-76). Numdam | MR 95g:82083 | Zbl 0798.35025 · Zbl 0798.35025
[4] F. BOUCHUT and L. DESVILLETTES , Averaging lemmas without time Fourier transform and applications to discretized kinetic equations . Preprint. · Zbl 0933.35159
[5] L. DESVILLETTES and S. MISCHLER , About the splitting algorithms for Boltzmann and BGK equations (Math. Model. Meth. Appl. Sc., Vol. 6, 1996 , pp. 1079-1101). MR 98d:82040 | Zbl 0876.35088 · Zbl 0876.35088
[6] R. J. DIPERNA and P.-L. LIONS , On the Cauchy problem for the Boltzmann Equation : global existence and weak stability results (Ann. Math., Vol. 130, 1989 , pp. 321-366). MR 90k:82045 | Zbl 0698.45010 · Zbl 0698.45010
[7] R. J. DIPERNA and P.-L. LIONS , Global weak solutions of Vlasov-Maxwell systems (Comm. Pure Appl. Math., Vol. 42, 1989 , pp. 729-757). MR 90i:35236 | Zbl 0698.35128 · Zbl 0698.35128
[8] R. J. DIPERNA , P.-L. LIONS and Y. MEYER , Lp regularity of velocity averages (Ann. Inst. H. Poincaré Ann. Non Linéaire, Vol. 8, 1991 , pp. 271-287). Numdam | MR 92g:35036 | Zbl 0763.35014 · Zbl 0763.35014
[9] P. GÉRARD , Moyennisation et régularité deux-microlocale , (Ann. Sci. Ecole Normale Sup., Vol. 4, 1990 , pp. 89-121). Numdam | MR 91g:35068 | Zbl 0725.35003 · Zbl 0725.35003
[10] P. GÉRARD , Microlocal defect measures (Comm. Partial Differential equations, Vol. 16(11), 1991 , pp. 1761-1794). MR 92k:35027 | Zbl 0770.35001 · Zbl 0770.35001
[11] F. GOLSE , B. PERTHAME and R. SENTIS Un résultat de compacité pour les équations du transport... (C. R. Acad. Sci. Paris, Série I, Vol. 301, 1985 , pp. 341-344). MR 86m:82062 | Zbl 0591.45007 · Zbl 0591.45007
[12] F. GOLSE , P.-L. LIONS , B. PERTHAME and R. SENTIS , Regularity of the moments of the solution of a transport equation (J. Funct. Anal., Vol. 76, 1988 , pp. 110-125). MR 89a:35179 | Zbl 0652.47031 · Zbl 0652.47031
[13] P.-L. LIONS , Régularité optimale des moyennes en vitesses (C. R. Acad. Sci. Paris, Série I, Vol. 320, 1995 , pp. 911-915). MR 96c:35184 | Zbl 0827.35110 · Zbl 0827.35110
[14] P. L. LIONS , B. PERTHAME and E. TADMOR , A kinetic formulation of multidimensional scalar conservation laws and related equations (J. of AMS, Vol. 7, 1994 , pp. 169-191). MR 94d:35100 | Zbl 0820.35094 · Zbl 0820.35094
[15] E. M. STEIN , Singular Integrals and Differentiability Properties of Functions , Princeton University Press, Princeton, 1970 . MR 44 #7280 | Zbl 0207.13501 · Zbl 0207.13501
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.