On decay of periodic entropy solutions to a scalar conservation law
Panov, E.Yu.
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 997-1007 / Harvested from Numdam

Nous considérons les lois de conservation [hyperboliques] en plusieurs dimensions dʼespace avec la fonction de flux seulement continue. Nous établissons une condition nécessaire et suffisante pour la décroissance des solutions entropiques périodiques de ce problème.

We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector.

Publié le : 2013-01-01
DOI : https://doi.org/10.1016/j.anihpc.2012.12.009
Classification:  35L65,  35B10,  35B35
@article{AIHPC_2013__30_6_997_0,
     author = {Panov, E.Yu.},
     title = {On decay of periodic entropy solutions to a scalar conservation law},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {997-1007},
     doi = {10.1016/j.anihpc.2012.12.009},
     mrnumber = {3132413},
     zbl = {1288.35347},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_6_997_0}
}
Panov, E.Yu. On decay of periodic entropy solutions to a scalar conservation law. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 997-1007. doi : 10.1016/j.anihpc.2012.12.009. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_6_997_0/

[1] G.-Q. Chen, H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch. Ration. Mech. Anal. 146 no. 2 (1999), 95-127 | MR 1718482 | Zbl 0942.35031

[2] R.J. Diperna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal. 88 (1985), 223-270 | MR 775191 | Zbl 0616.35055

[3] P. Gerárd, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), 1761-1794 | MR 1135919 | Zbl 0770.35001

[4] S.N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sb. 81 (1970), 228-255, Math. USSR-Sb. 10 (1970), 217-243 | MR 267257 | Zbl 0215.16203

[5] S.N. Kruzhkov, E.Yu. Panov, First-order conservative quasilinear laws with an infinite domain of dependence on the initial data, Dokl. Akad. Nauk SSSR 314 (1990), 79-84, Soviet Math. Dokl. 42 (1991), 316-321 | MR 1118483 | Zbl 0789.35039

[6] S.N. Kruzhkov, E.Yu. Panov, Osgoodʼs type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order, Ann. Univ. Ferrara Sez. VII (N.S.) 40 (1994), 31-54 | MR 1399621 | Zbl 0863.35060

[7] P.V. Lysuho, E.Yu. Panov, Renormalized entropy solutions to the Cauchy problem for first order quasilinear conservation laws in the class of periodic functions, J. Math. Sci. 177 no. 1 (2011), 27-49 | MR 2838985 | Zbl 1290.35152

[8] E.Yu. Panov, On sequences of measure-valued solutions of first-order quasilinear equations, Mat. Sb. 185 no. 2 (1994), 87-106, Russian Acad. Sci. Sb. Math. 81 no. 1 (1995), 211-227 | MR 1264775 | Zbl 0924.35026

[9] E.Yu. Panov, On strong precompactness of bounded sets of measure valued solutions for a first order quasilinear equation, Mat. Sb. 186 no. 5 (1995), 103-114, Russian Acad. Sci. Sb. Math. 186 no. 5 (1995), 729-740 | MR 1341087 | Zbl 0839.35139

[10] E.Yu. Panov, Property of strong precompactness for bounded sets of measure valued solutions of a first-order quasilinear equation, Mat. Sb. 190 no. 3 (1999), 109-128, Russian Acad. Sci. Sb. Math. 190 no. 3 (1999), 427-446 | MR 1700996 | Zbl 0936.35047

[11] E.Yu. Panov, A remark on the theory of generalized entropy sub- and supersolutions of the Cauchy problem for a first-order quasilinear equation, Differ. Uravn. 37 no. 2 (2001), 252-259, Differ. Equ. 37 no. 2 (2001), 272-280 | MR 1849443 | Zbl 1001.35024

[12] E.Yu. Panov, Maximum and minimum generalized entropy solutions to the Cauchy problem for a first-order quasilinear equation, Mat. Sb. 193 no. 5 (2002), 95-112, Russian Acad. Sci. Sb. Math. 193 no. 5 (2002), 727-743 | MR 1918249 | Zbl 1058.35154

[13] E.Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ. 2 no. 4 (2005), 885-908 | MR 2195985 | Zbl 1145.35429

[14] E.Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ. 4 no. 4 (2007), 729-770 | MR 2374223 | Zbl 1144.35037

[15] E.Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal. 195 no. 2 (2010), 643-673 | MR 2592291 | Zbl 1191.35102

[16] 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 584398 | Zbl 0437.35004

[17] 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 1069518 | Zbl 0774.35008