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Regularity in kinetic formulations via averaging lemmas

Published online by Cambridge University Press:  15 August 2002

Pierre-Emmanuel Jabin  and
Benoît Perthame
Pierre-Emmanuel Jabin
Affiliation:
École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553 du CNRS, 45 rue d'Ulm, 75230 Paris Cedex 05, France; jabin@dma.ens.fr.
Benoît Perthame
Affiliation:
École Normale Supérieure, Département de Mathématiques et Applications, UMR 8553 du CNRS, 45 rue d'Ulm, 75230 Paris Cedex 05, France; perthame@dma.ens.fr.
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Abstract

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.

Keywords

Regularizing effectskinetic formulationaveraging lemmashyperbolic equationsline-energy Ginzburg–Landau.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 761 - 774
Copyright
© EDP Sciences, SMAI, 2002

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References

Ambrosio, L., De Lellis, C. and Mantegazza, C., Line energies for gradient vector fields in the plane. Calc. Var. Partial Differential Equations 9 (1999) 327-355. CrossRef
J. Bergh and J. Löfström, Interpolation spaces, an introduction. Springer-Verlag, A Ser. of Comprehensive Stud. in Math. 223 (1976).
Brenier, Y. and Corrias, L., A kinetic formulation formulti-branch entropy solutions of scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 169-190. CrossRef
Bézard, M., Régularité L p précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122 (1994) 29-76. CrossRef
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.
F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theory. Gauthiers-Villars, Ser. in Appl. Math. (2000).
A. Desimone, R.W. Kohn, S. Müller and F. Otto, Magnetic microstructures, a paradigm of multiscale problems. Proc. of ICIAM (to appear).
DeVore, R. and Petrova, G.P., The averaging lemma. J. Amer. Math. Soc. 14 (2001) 279-296. CrossRef
DiPerna, R. and Lions, P.L., Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42 (1989) 729-757. CrossRef
DiPerna, R., Lions, P.L. and Meyer, Y., L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 271-287. CrossRef
Gérard, P., Microlocal defect measures. Comm. Partial Differential Equations 16 (1991) 1761-1794. CrossRef
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.
Golse, F., Lions, P.L., Perthame, B. and Sentis, R., Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 26 (1988) 110-125. CrossRef
Golse, F., Perthame, B. and Sentis, R., 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.
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).
Jabin, P.-E. and Perthame, B., Compactness in Ginzburg-Landau energy by kinetic averaging. Comm. Pure Appl. Math. 54 (2001) 1096-1109. CrossRef
P.-E. Jabin, F. Otto and B. Perthame, Line-energy Ginzburg-Landau models: Zero-energy states. Ann. Sc. Norm. Sup. Pisa (to appear).
Lions, J.-L. and Peetre, J., Sur une classe d'espaces d'interpolation. Inst. Hautes Études Sci. Publ. Math. 19 (1964) 5-68. CrossRef
Lions, P.L., Régularité optimale des moyennes en vitesse. C. R. Acad. Sci. Sér. I Math. 320 (1995) 911-915.
Lions, P.L., Perthame, B. and Tadmor, E., A kinetic formulation of multidimensional scalar conservation laws and related questions. J. Amer. Math. Soc. 7 (1994) 169-191. CrossRef
Lions, P.L., Perthame, B. and Tadmor, E., Kinetic formulation of the isentropic gas dynamics and p-systems. Comm. Math. Phys. 163 (1994) 415-431. CrossRef
Ole, O.A. {\u{\i}}\kern.15em nik, On Cauchy's problem for nonlinear equations in a class of discontinuous functions. Doklady Akad. Nauk SSSR (N.S.) 95 (1954) 451-454.
B. Perthame, Kinetic Formulations of conservation laws. Oxford University Press, Oxford Ser. in Math. and Its Appl. (2002).
Perthame, B. and Souganidis, P.E., A limiting case for velocity averaging. Ann. Sci. École Norm. Sup. (4) 31 (1998) 591-598. CrossRef
M. Porthileiro, Compactness of velocity averages. Preprint.
T. Rivière and S. Serfaty, Compactness, kinetic formulation, and entropies for a problem related to micromagnetics. Preprint (2001).
Vasseur, A., Time regularity for the system of isentropic gas dynamics with γ = 3. Comm. Partial Differential Equations 24 (1999) 1987-1997. CrossRef
M. Westdickenberg, some new velocity averaging results. SIAM J. Math. Anal. (to appear).
Cheverry, C., Regularizing effects for multidimensional scalar conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000) 413-472. CrossRef