Regularity in kinetic formulations via averaging lemmas
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 761-774.

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 γ=3 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 K-method of real interpolation.

DOI: 10.1051/cocv:2002033
Classification: 35L65,  35B30,  35B65,  74G65,  83D30
Keywords: regularizing effects, kinetic formulation, averaging lemmas, hyperbolic equations, line-energy Ginzburg-Landau
     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},
     zbl = {1065.35185},
     mrnumber = {1932972},
     language = {en},
     url = {}
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
DA  - 2002///
SP  - 761
EP  - 774
VL  - 8
PB  - EDP-Sciences
UR  -
UR  -
UR  -
UR  -
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
%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, Volume 8 (2002), pp. 761-774. doi : 10.1051/cocv:2002033.

[1] L. Ambrosio, C. De Lellis and C. Mantegazza, Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9 (1999) 327-355. | MR | Zbl

[2] J. Bergh and J. Löfström, Interpolation spaces, an introduction. Springer-Verlag, A Ser. of Comprehensive Stud. in Math. 223 (1976). | MR | Zbl

[3] Y. Brenier and L. Corrias, 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

[4] M. Bézard, Régularité L p précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122 (1994) 29-76. | EuDML | Numdam | MR | Zbl

[5] F. Bouchut and L. Desvillettes, Averaging lemmas without time Fourier transform and applications to discretized kineticequations. Proc. Roy. Soc. Edinburgh Ser. A 129 (1999) 19-36. | MR | Zbl

[6] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theory. Gauthiers-Villars, Ser. in Appl. Math. (2000). | MR | Zbl

[7] A. Desimone, R.W. Kohn, S. Müller and F. Otto, Magnetic microstructures, a paradigm of multiscale problems. Proc. of ICIAM (to appear). | Zbl

[8] R. Devore and G.P. Petrova, The averaging lemma. J. Amer. Math. Soc. 14 (2001) 279-296. | MR | Zbl

[9] R. Diperna and P.L. Lions, Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42 (1989) 729-757. | Zbl

[10] R. Diperna, P.L. Lions and Y. Meyer, L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 271-287. | Numdam | MR | Zbl

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

[12] F. Golse, 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] F. Golse, P.L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 26 (1988) 110-125. | MR | Zbl

[14] F. Golse, B. Perthame and R. Sentis, 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

[15] S. Hwang and A. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Preprint. University of Wisconsin, Madison (2001). | MR | Zbl

[16] P.-E. Jabin and B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. | Zbl

[17] P.-E. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg-Landau models: Zero-energy states. Ann. Sc. Norm. Sup. Pisa (to appear). | Numdam | Zbl

[18] J.-L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation. Inst. Hautes Études Sci. Publ. Math. 19 (1964) 5-68. | Numdam | Zbl

[19] P.L. Lions, Régularité optimale des moyennes en vitesse. C. R. Acad. Sci. Sér. I Math. 320 (1995) 911-915. | MR | Zbl

[20] P.L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related questions. J. Amer. Math. Soc. 7 (1994) 169-191. | MR | Zbl

[21] P.L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and p-systems. Comm. Math. Phys. 163 (1994) 415-431. | MR | Zbl

[22] O.A. Oleĭnik, 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] B. Perthame, Kinetic Formulations of conservation laws. Oxford University Press, Oxford Ser. in Math. and Its Appl. (2002). | MR | Zbl

[24] B. Perthame and P.E. Souganidis, A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4) 31 (1998) 591-598. | Numdam | MR | Zbl

[25] M. Porthileiro, Compactness of velocity averages. Preprint. | Zbl

[26] T. Rivière and S. Serfaty, Compactness, kinetic formulation, and entropies for a problem related to micromagnetics. Preprint (2001). | MR | Zbl

[27] A. Vasseur, Time regularity for the system of isentropic gas dynamics with γ=3. Comm. Partial Differential Equations 24 (1999) 1987-1997. | MR | Zbl

[28] M. Westdickenberg, some new velocity averaging results. SIAM J. Math. Anal. (to appear). | MR | Zbl

[29] C. Cheverry, Regularizing effects for multidimensional scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 413-472. | Numdam | MR | Zbl

Cited by Sources: