We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity regularity for the solution to the transport equation under consideration. The method of proof is based on a decomposition of the density in Fourier space, combined with the -method of real interpolation.
Mots clés : regularizing effects, kinetic formulation, averaging lemmas, hyperbolic equations, line-energy Ginzburg-Landau
@article{COCV_2002__8__761_0, author = {Jabin, Pierre-Emmanuel and Perthame, Beno{\^\i}t}, title = {Regularity in kinetic formulations via averaging lemmas}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {761--774}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, doi = {10.1051/cocv:2002033}, mrnumber = {1932972}, zbl = {1065.35185}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2002033/} }
TY - JOUR AU - Jabin, Pierre-Emmanuel AU - Perthame, Benoît TI - Regularity in kinetic formulations via averaging lemmas JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 SP - 761 EP - 774 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002033/ DO - 10.1051/cocv:2002033 LA - en ID - COCV_2002__8__761_0 ER -
%0 Journal Article %A Jabin, Pierre-Emmanuel %A Perthame, Benoît %T Regularity in kinetic formulations via averaging lemmas %J ESAIM: Control, Optimisation and Calculus of Variations %D 2002 %P 761-774 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2002033/ %R 10.1051/cocv:2002033 %G en %F COCV_2002__8__761_0
Jabin, Pierre-Emmanuel; Perthame, Benoît. Regularity in kinetic formulations via averaging lemmas. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 761-774. doi : 10.1051/cocv:2002033. http://www.numdam.org/articles/10.1051/cocv:2002033/
[1] Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9 (1999) 327-355. | MR | Zbl
, and ,[2] Interpolation spaces, an introduction. Springer-Verlag, A Ser. of Comprehensive Stud. in Math. 223 (1976). | MR | Zbl
and ,[3] A kinetic formulation formulti-branch entropy solutions of scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 169-190. | EuDML | Numdam | MR | Zbl
and ,[4] Régularité précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122 (1994) 29-76. | EuDML | Numdam | MR | Zbl
,[5] Averaging lemmas without time Fourier transform and applications to discretized kineticequations. Proc. Roy. Soc. Edinburgh Ser. A 129 (1999) 19-36. | MR | Zbl
and ,[6] Kinetic equations and asymptotic theory. Gauthiers-Villars, Ser. in Appl. Math. (2000). | MR | Zbl
, and ,[7] Magnetic microstructures, a paradigm of multiscale problems. Proc. of ICIAM (to appear). | Zbl
, , and ,[8] The averaging lemma. J. Amer. Math. Soc. 14 (2001) 279-296. | MR | Zbl
and ,[9] Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42 (1989) 729-757. | Zbl
and ,[10] regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 271-287. | Numdam | MR | Zbl
, and ,[11] Microlocal defect measures. Comm. Partial Differential Equations 16 (1991) 1761-1794. | MR | Zbl
,[12] Quelques résultats de moyennisation pour les équations aux dérivées partielles. Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo Speciale 1988 Hyperbolic equations (1987) 101-123. | MR | Zbl
,[13] Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 26 (1988) 110-125. | MR | Zbl
, , and ,[14] Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 341-344. | Zbl
, and ,[15] Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Preprint. University of Wisconsin, Madison (2001). | MR | Zbl
and ,[16] Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. | Zbl
and ,[17] Line-energy Ginzburg-Landau models: Zero-energy states. Ann. Sc. Norm. Sup. Pisa (to appear). | Numdam | Zbl
, and ,[18] Sur une classe d'espaces d'interpolation. Inst. Hautes Études Sci. Publ. Math. 19 (1964) 5-68. | Numdam | Zbl
and ,[19] Régularité optimale des moyennes en vitesse. C. R. Acad. Sci. Sér. I Math. 320 (1995) 911-915. | MR | Zbl
,[20] A kinetic formulation of multidimensional scalar conservation laws and related questions. J. Amer. Math. Soc. 7 (1994) 169-191. | MR | Zbl
, and ,[21] Kinetic formulation of the isentropic gas dynamics and -systems. Comm. Math. Phys. 163 (1994) 415-431. | MR | Zbl
, and ,[22] On Cauchy's problem for nonlinear equations in a class of discontinuous functions. Doklady Akad. Nauk SSSR (N.S.) 95 (1954) 451-454. | Zbl
,[23] Kinetic Formulations of conservation laws. Oxford University Press, Oxford Ser. in Math. and Its Appl. (2002). | MR | Zbl
,[24] A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4) 31 (1998) 591-598. | Numdam | MR | Zbl
and ,[25] Compactness of velocity averages. Preprint. | Zbl
,[26] Compactness, kinetic formulation, and entropies for a problem related to micromagnetics. Preprint (2001). | MR | Zbl
and ,[27] Time regularity for the system of isentropic gas dynamics with . Comm. Partial Differential Equations 24 (1999) 1987-1997. | MR | Zbl
, , some new velocity averaging results. SIAM J. Math. Anal. (to appear). |[29] Regularizing effects for multidimensional scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 413-472. | Numdam | MR | Zbl
,Cité par Sources :