Ultra-parabolic H-measures and compensated compactness
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, pp. 47-62.

We present a generalization of compensated compactness theory to the case of variable and generally discontinuous coefficients, both in the quadratic form and in the linear, up to the second order, constraints. The main tool is the localization properties for ultra-parabolic H-measures corresponding to weakly convergent sequences.

Nous présentons ici une généralisation de la théorie de la « compacité par compensation ». Le cas d'une forme quadratique et de contraintes différentielles avec coefficients variables, éventuellement discontinus en espace, est considéré. Ces contraintes différentielles peuvent être d'ordre un, mais aussi d'ordre deux. Notre outil principal est le principe de localisation pour les H-mesures ultra-paraboliques associées à des suites de fonctions faiblement convergentes.

DOI: 10.1016/j.anihpc.2010.10.002
Classification: 35A27, 35K55, 46G10, 42B15, 42B30
Keywords: Ultra-parabolic H-measures, Localization principles, Compensated compactness, Measure valued functions, Semi-linear parabolic equations
@article{AIHPC_2011__28_1_47_0,
     author = {Panov, E.Yu.},
     title = {Ultra-parabolic {\protect\emph{H}-measures} and compensated compactness},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {47--62},
     publisher = {Elsevier},
     volume = {28},
     number = {1},
     year = {2011},
     doi = {10.1016/j.anihpc.2010.10.002},
     mrnumber = {2765509},
     zbl = {1211.35013},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2010.10.002/}
}
TY  - JOUR
AU  - Panov, E.Yu.
TI  - Ultra-parabolic H-measures and compensated compactness
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2011
SP  - 47
EP  - 62
VL  - 28
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2010.10.002/
DO  - 10.1016/j.anihpc.2010.10.002
LA  - en
ID  - AIHPC_2011__28_1_47_0
ER  - 
%0 Journal Article
%A Panov, E.Yu.
%T Ultra-parabolic H-measures and compensated compactness
%J Annales de l'I.H.P. Analyse non linéaire
%D 2011
%P 47-62
%V 28
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2010.10.002/
%R 10.1016/j.anihpc.2010.10.002
%G en
%F AIHPC_2011__28_1_47_0
Panov, E.Yu. Ultra-parabolic H-measures and compensated compactness. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 1, pp. 47-62. doi : 10.1016/j.anihpc.2010.10.002. http://www.numdam.org/articles/10.1016/j.anihpc.2010.10.002/

[1] N. Antonić, M. Lazar, H-measures and variants applied to parabolic equations, J. Math. Anal. Appl. 343 (2008), 207-225 | MR | Zbl

[2] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin (1980) | MR

[3] G.A. Francfort, An introduction to H-measures and their applications, G. Dal Maso, A. Desimone, Antonio F. Tomarelli (ed.), Variational Problems in Materials Science, Progr. Nonlinear Differential Equations Appl. vol. 68, Birkhäuser Verlag, Basel (2006), 85-110

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

[5] F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa 5 (1978), 489-507 | EuDML | Numdam | MR | Zbl

[6] E.Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property, J. Math. Sci. 159 no. 2 (2009), 180-228 | MR | Zbl

[7] E.Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux, J. Differential Equations 247 no. 10 (2009), 2821-2870 | MR | Zbl

[8] P. Pedregal, Parametrized Measures and Variational Principles, Progr. Nonlinear Differential Equations Appl. vol. 30, Birkhäuser Verlag, Basel (1997) | MR | Zbl

[9] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. vol. 30, Princeton University Press, Princeton, NJ (1970) | MR | Zbl

[10] L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot. Watt Symposium, vol. 4, Edinburgh, 1979, Res. Notes Math. vol. 39 (1979), 136-212 | MR | Zbl

[11] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 no. 3–4 (1990), 193-230 | MR | Zbl

Cited by Sources: