Discontinuous travelling wave solutions for a class of dissipative hyperbolic models
Currò, Carmela ; Fusco, Domenico
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 61-71 / Harvested from Biblioteca Digitale Italiana di Matematica

Discontinuous shock structure solutions for a general system of balance laws is considered in order to investigate the problem of connecting two equilibrium states lying on different sides of a singular barrier representing a locus of irregular singular points for travelling waves. Within such a theoretical setting a governing system of monoatomic gas is considered.

Si considerano le soluzioni di tipo struttura d'urto per un sistema di equazioni di bilancio allo scopo di studiare la connessione tra due stati di equilibrio separati nello spazio delle fasi da una barriera singolare, rappresentante un luogo di punti di singolarità nello studio delle «travelling waves». Si considerano infine le equazioni che descrivono il bilancio di un gas monoatomico uni-dimensionale dedotte nell'ambito della Termodinamica Estesa.

Publié le : 2005-03-01
@article{RLIN_2005_9_16_1_61_0,
     author = {Carmela Curr\`o and Domenico Fusco},
     title = {Discontinuous travelling wave solutions for a class of dissipative hyperbolic models},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {16},
     year = {2005},
     pages = {61-71},
     zbl = {1225.35144},
     mrnumber = {2225923},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2005_9_16_1_61_0}
}
Currò, Carmela; Fusco, Domenico. Discontinuous travelling wave solutions for a class of dissipative hyperbolic models. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005) pp. 61-71. http://gdmltest.u-ga.fr/item/RLIN_2005_9_16_1_61_0/

[1] Müller, I. - Ruggeri, T., Extended Thermodynamics. Springer Tracts on Natural Philosophy, 37, Springer-Verlag, New York 1993, 231 pp. (Rational Extended Thermodynamics. Springer-Verlag, new edition, 37, 1998, 393 pp.). | MR 1632151 | Zbl 0895.00005

[2] Boillat, G. - Ruggeri, T., Hyperbolic Principal Subsystems: Entropy Convexity and Subcharacteristic Conditions. Arch. Rat. Mech. Anal., 137, 1997, 304-320. | MR 1463797 | Zbl 0878.35070

[3] Godunov, S.K., An interesting class of quasilinear systems. Sov. Math., 2, 1961, 947-948. | MR 116141 | Zbl 0125.06002

[4] Friedrichs, K.O. - Lax, P.D., Systems of conservation equation with a convex extension. Proc. Nat. Acad. Sci. USA, 68, 1971, 1686-1688. | MR 285799 | Zbl 0229.35061

[5] Boillat, G., Sur l'existence et la recherche d'équations de conservation supplémentaires pour les systémes hyperboliques. C.R. Acad. Sciences Paris, 278 A, 1974, 909-912. | MR 342870 | Zbl 0279.35058

[6] Boillat, G., Non Linear Fields and Waves. In: T. Ruggeri (ed.), Recent Mathematical Methods in Nonlinear Wave Propagation. CIME Course (Montecatini 1994). Lecture Notes in Mathematics, 1640, Springer-Verlag, 1996, 103-152. | MR 1600900 | Zbl 0877.35080

[7] Ruggeri, T. - Strumia, A., Main field and convex covariant density for quasi-linear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré, 34 A, 1981, 65-84. | MR 605357 | Zbl 0473.76126

[8] Whitham, G.B., Linear and Nonlinear Waves. J. Wiley-Interscience, New York1974, 636 pp. | MR 483954 | Zbl 0373.76001

[9] Ruggeri, T., Breakdown of shock-wave-structure solutions. Phys. Rev., E47(6), 1993, 4135-4140. | MR 1377905

[10] Boillat, G. - Ruggeri, T., On the shock structure problem for hyperbolic system of balance laws and convex entropy. Continuum Mech. Thermodyn., 10, 1998, 285-292. | MR 1652858 | Zbl 0922.76237

[11] Weiss, W., Die Berechnung von kontinuierlichen Stofstrukturen in der Kinetischen Gastheorie. Habilitation thesis TU, Berlin1997.

[12] Kamke, E., Differentialgleichungen I. Geest & Portig, Leipzig1967. | Zbl 0194.39301

[13] Bieberbach, L., Theorie der gewohnlichen Differentialgleichungen. Springer-Verlag, Berlin1965, 389 pp. | JFM 50.0639.02 | MR 176133

[14] Marchant, B.P. - Norbury, J. - Perumpanani, A.J., Traveling shock waves arising in a model of malignant invasion. SIAM J. Appl. Math., 60, 2000, 463-476. | MR 1740255 | Zbl 0944.34021

[15] Marchant, B.P. - Norbury, J. - Sherratt, J.A., Discontinuous travelling wave solutions to a haptotaxis-dominated model of malignant invasion. Nonlinearity, 14, 2001, 1653-1671. | MR 1867097 | Zbl 0985.92012

[16] Marchant, B.P. - Norbury, J., Discontinuous travelling wave solutions for certain hyperbolic systems. IMA J. Appl. Math., 67, 2002, 201-224. | MR 1897286 | Zbl 1076.35071

[17] Pettet, G.J. - Mcelwain, D.L.S. - Norbury, J., Lotka-Volterra equations with chemotaxis: walls, barriers and travelling waves. IMA J. Appl. Med. Biol., 17, 2000, 395-413. | Zbl 0969.92020

[18] Lax, P.D., Shock Waves and Entropy. In: E. Zarantonello (ed.), Contribution to Non Linear Functional Analysis. Acad. Press, New York 1971, 603-634. | MR 393870 | Zbl 0268.35014

[19] Ruggeri, T., Shock waves in hyperbolic dissipative systems: non equilibrium gases. In: D. Fusco - A. Jeffrey (eds.), Euromech colloquium 270 (Reggio Calabria, 25-28 September 1990). Pitman Research Notes in Mathematics, vol. 227, Longman, Harlow 1991. | Zbl 0743.76049

[20] Curró, C. - Fusco, D., Shock-like travelling wave solutions for a hyperbolic tumour growth model. In: Proceedings of XII International Conference on Waves and Stability in Continuous Media (Villasimius, Cagliari, June 1-7 2003). World Scientific, 2003, 141-147. | MR 2089842 | Zbl 1069.35042