Geodesic
Long Time Asymptotics of Periodic Generalized Entropy Solutions of Scalar Conservation Laws
Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 133-143.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove that the periodic generalized entropy solution of a one-dimensional conservation law converges in time to a traveling wave. In this case, the flow function is linear on the minimal interval containing the essential image of the traveling wave profile and the wave velocity coincides with the angular coefficient of the flow function bounded on this interval.
Keywords: conservation law, generalized entropy solution, stabilization property, traveling wave, measure-valued function, compensated compactness. 
@article{MZM_2016_100_1_a9,
     author = {E. Yu. Panov},
     title = {Long {Time} {Asymptotics} of {Periodic} {Generalized} {Entropy} {Solutions} of {Scalar} {Conservation} {Laws}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {133--143},
     publisher = {mathdoc},
     volume = {100},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a9/}
}
TY  - JOUR
AU  - E. Yu. Panov
TI  - Long Time Asymptotics of Periodic Generalized Entropy Solutions of Scalar Conservation Laws
JO  - Matematičeskie zametki
PY  - 2016
SP  - 133
EP  - 143
VL  - 100
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a9/
LA  - ru
ID  - MZM_2016_100_1_a9
ER  - 
%0 Journal Article
%A E. Yu. Panov
%T Long Time Asymptotics of Periodic Generalized Entropy Solutions of Scalar Conservation Laws
%J Matematičeskie zametki
%D 2016
%P 133-143
%V 100
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a9/
%G ru
%F MZM_2016_100_1_a9
E. Yu. Panov. Long Time Asymptotics of Periodic Generalized Entropy Solutions of Scalar Conservation Laws. Matematičeskie zametki, Tome 100 (2016) no. 1, pp. 133-143. http://geodesic.mathdoc.fr/item/MZM_2016_100_1_a9/

[1] S. N. Kruzhkov, “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Matem. sb., 81 (123):2 (1970), 228–255 | MR | Zbl

[2] E. Yu. Panov, “Meroznachnye resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka s neogranichennoi oblastyu zavisimosti ot nachalnykh dannykh”, Dinamika sploshnoi sredy, 88 (1988), 102–108 | Zbl

[3] S. N. Kruzhkov, E. Yu. Panov, “Konservativnye kvazilineinye zakony pervogo poryadka s beskonechnoi oblastyu zavisimosti ot nachalnykh dannykh”, Dokl. AN SSSR, 314:1 (1990), 79–84 | MR | Zbl

[4] S. N. Kruzhkov, E. Yu. Panov, “Osgood's 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 | Zbl

[5] E. Yu. Panov, “O edinstvennosti resheniya zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka s odnoi dopustimoi strogo vypukloi entropiei”, Matem. zametki, 55:5 (1994), 116–129 | MR | Zbl

[6] E. Yu. Panov, “K teorii obobschennykh entropiinykh sub- i super-reshenii zadachi Koshi dlya kvazilineinogo uravneniya pervogo poryadka”, Differents. uravneniya, 37:2 (2001), 252–259 | MR | Zbl

[7] E. Yu. Panov, “On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws”, Netw. Heterog. Media, 11:2 (2016), 349–367 | MR

[8] L. Tartar, “Compensated compactness and applications to partial differential equations.”, Nonlinear Analysis and Mechanics, Heriot–Watt Symposium, Vol. IV, Res. Notes in Math., 39, Pitman, Boston, MA, 1979, 136–212 | MR | Zbl

[9] G. Q. Chen, Y. G. Lu, “The study on application way of the compensated compactness theory”, Chinese Sci. Bull., 34:1 (1989), 15–19 | MR | Zbl

[10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wiss., 325, Springer-Verlag, Berlin, 2010 | MR | Zbl

[11] E. Panov, “On weak completeness of the set of entropy solutions to a scalar conservation law”, SIAM J. Math. Anal., 41:1 (2009), 26–36 | DOI | MR | Zbl

[12] P. V. Lysukho, E. Yu. Panov, “Renormalizovannye entropiinye resheniya zadachi Koshi dlya kvazilineinykh zakonov sokhraneniya pervogo poryadka v klasse periodicheskikh funktsii”, Probl. matem. anal., 59 (2011), 25–42

[13] G.-Q. Chen, H. Frid, “Decay of entropy solutions of nonlinear conservation laws”, Arch. Ration. Mech. Anal., 146:2 (1999), 95–127 | DOI | MR | Zbl

[14] C. M. Dafermos, “Long time behavior of periodic solutions to scalar conservation laws in several space dimensions”, SIAM J. Math. Anal., 45:4 (2013), 2064–2070 | DOI | MR | Zbl

[15] R. J. DiPerna, “Measure-valued solutions to conservation laws”, Arch. Ration. Mech. Anal., 88:3 (1985), 223–270 | DOI | MR | Zbl

[16] E. Panov, “On weak completeness of the set of entropy solutions to a degenerate nonlinear parabolic equation”, SIAM J. Math. Anal., 44:1 (2012), 513–535 | DOI | MR | Zbl

[17] F. Murat, “L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q2$”, J. Math. Pures Appl. (9), 60:3 (1981), 309–322 | MR | Zbl

[18] F. Murat, “Compacité par compensation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5:3 (1978), 489–507 | MR | Zbl

[19] E. Yu. Panov, “Existence of strong traces for generalized solutions of multidimensional scalar conservation laws”, J. Hyperbolic Differ. Equ., 2:4 (2005), 885–908 | DOI | MR | Zbl