## Wave front tracking in systems of conservation laws.(English)Zbl 1099.35063

Summary: This paper contains several recent results about nonlinear systems of hyperbolic conservation laws obtained through the technique of wave front tracking.

### MSC:

 35L65 Hyperbolic conservation laws 35A35 Theoretical approximation in context of PDEs 35B35 Stability in context of PDEs 35D05 Existence of generalized solutions of PDE (MSC2000)

### Keywords:

conservation laws; wave front tracking
Full Text:

### References:

 [1] R. Abeyaratne, J. K. Knowles: Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational Mech. Anal. 114 (1991), 119-154. · Zbl 0745.73001 [2] D. Amadori: Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA Nonlinear Differential Equations Appl. 4 (1997), 1-42. · Zbl 0868.35069 [3] D. Amadori, R. M. Colombo: Continuous dependence for $$2\times 2$$ conservation laws with boundary. J. Differential Equations 138 (1997), 229-266. · Zbl 0884.35091 [4] D. Amadori, R. M. Colombo: Viscosity solutions and standard Riemann semigroup for conservation laws with boundary. Rend. Sem. Mat. Univ. Padova 99 (1998), 219-245. · Zbl 0910.35078 [5] D. Amadori, L. Gosse, and G. Guerra: Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws. Arch. Rational Mech. Anal. 162 (2002), 327-366. · Zbl 1024.35066 [6] D. Amadori, G. Guerra: Uniqueness and continuous dependence for systems of balance laws with dissipation. Nonlinear Anal. 49A (2002), 987-1014. · Zbl 1006.35062 [7] F. Ancona, G. M. Coclite: On the attainable set for Temple class systems with boundary controls. Preprint (2002). [8] F. Ancona, A. Marson: On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36 (1998), 290-312. · Zbl 0919.35082 [9] F. Ancona, A. Marson: $$L^1$$-stability for $$n\times n$$ non genuinely nonlinear conservation laws. Preprint (2000). [10] F. Ancona, A. Marson: A wavefront tracking algorithm for $$N\times N$$ nongenuinely nonlinear conservation laws. J. Differential Equations 177 (2001), 454-493. · Zbl 0993.35062 [11] F. Asakura: Kinetic condition and the Gibbs function. Taiwanese J. Math. 4 (2000), 105-117. · Zbl 0951.35078 [12] P. Baiti, A. Bressan: The semigroup generated by a Temple class system with large data. Differential Integral Equations 10 (1997), 401-418. · Zbl 0890.35083 [13] P. Baiti, H. K. Jenssen: On the front-tracking algorithm. J. Math. Anal. Appl. 217 (1998), 395-404. · Zbl 0966.35078 [14] P. Baiti and H. K. Jenssen: Blowup in $$L^\infty$$ for a class of genuinely nonlinear hyperbolic systems of conservation laws. Discrete Contin. Dynam. Systems 7 (2001), 837-853. · Zbl 1200.35185 [15] S. Bianchini: The semigroup generated by a Temple class system with non-convex flux function. Differential Integral Equations 13 (2000), 1529-1550. · Zbl 1043.35110 [16] S. Bianchini, A. Bressan: Vanishing viscosity solutions of nonlinear hyperbolic systems. Annals of Mathematics · Zbl 1082.35095 [17] S. Bianchini, R. M. Colombo: On the stability of the standard Riemann semigroup. Proc. Amer. Math. Soc. 130 (2002), 1961-1973 · Zbl 1026.35073 [18] A. Bressan: Global solutions of systems of conservation laws by wave-front tracking. J. Math. Anal. Appl. 170 (1992), 414-432. · Zbl 0779.35067 [19] A. Bressan: The unique limit of the Glimm scheme. Arch. Rational Mech. Anal. 130 (1995), 205-230. · Zbl 0835.35088 [20] A. Bressan: The semigroup approach to systems of conservation laws. Fourth Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 1995). Math. Contemp. 10 (1996), 21-74. [21] A. Bressan: Uniqueness and stability of weak solutions to systems of conservation laws. Proceedings of the 9th International Conference on Waves and Stability in Continuous Media (Bari, 1997), Suppl. Rend. Circ. Mat. Palermo (2), Ser. 57, 1998, pp. 69-78. · Zbl 0941.35048 [22] A. Bressan: Hyperbolic Systems of Conservation Laws. Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000. · Zbl 1157.35421 [23] A. Bressan: The front tracking method for systems of conservation laws. Handbook of Partial Differential Equations, C. M. Dafermos, and E. Feireisl (eds.), Elsevier. To appear. · Zbl 1074.35001 [24] A. Bressan, G. M. Coclite: On the boundary control of systems of conservation laws. SIAM J. Control Optim. 41 (2002), 607-622 · Zbl 1026.35075 [25] A. Bressan, R. M. Colombo: The semigroup generated by $$2\times 2$$ conservation laws. Arch. Rational Mech. Anal. 133 (1995), 1-75. · Zbl 0849.35068 [26] A. Bressan, R. M. Colombo: Unique solutions of $$2\times 2$$ conservation laws with large data. Indiana Univ. Math. J. 44 (1995), 677-725. · Zbl 0852.35092 [27] A. Bressan, R. M. Colombo: Decay of positive waves in nonlinear systems of conservation laws. Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV 26 (1998), 133-160. · Zbl 0906.35059 [28] A. Bressan, G. Crasta, and B. Piccoli: Well-posedness of the Cauchy problem for $$n\times n$$ systems of conservation laws. Mem. Amer. Math. Soc. 146(694), 2000. · Zbl 0958.35001 [29] A. Bressan, P. Goatin: Oleinik type estimates and uniqueness for $$n\times n$$ conservation laws. J. Differential Equations 156 (1999), 26-49. · Zbl 0990.35095 [30] A. Bressan, G. Guerra: Shift-differentiability of the flow generated by a conservation law. Discrete Contin. Dynam. Systems 3 (1997), 35-58. · Zbl 0948.35077 [31] A. Bressan, P. G. LeFloch: Uniqueness of weak solutions to systems of conservation laws. Arch. Rational Mech. Anal. 140 (1997), 301-317. · Zbl 0903.35039 [32] A. Bressan, P. G. LeFloch: Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws. Indiana Univ. Math. J. 48 (1999), 43-84. · Zbl 0932.35139 [33] A. Bressan, M. Lewicka: Shift differentials of maps in BV spaces. Nonlinear Theory of Generalized Function (Vienna, 1997), Vol. 401 of Chapman & Hall/CRC Res. Notes Math, 401, Chapman & Hall/CRC, Boca Raton, 1999, pp. 47-61. · Zbl 0935.46025 [34] A. Bressan, T.-P. Liu, and T. Yang: $$L^1$$ stability estimates for $$n\times n$$ conservation laws. Arch. Rational Mech. Anal. 149 (1999), 1-22. · Zbl 0938.35093 [35] R. M. Colombo: Hyperbolic phase transitions in traffic flow. SIAM J. Appl. Math. 63 (2002), 708-721. · Zbl 1037.35043 [36] R. M. Colombo, A. Corli: Continuous dependence in conservation laws with phase transitions. SIAM J. Math. Anal. 31 (1999), 34-62. · Zbl 0977.35085 [37] R. M. Colombo, A. Corli: Sonic hyperbolic phase transitions and Chapman-Jouguet detonations. J. Differential Equations 184 (2002), 321-347. · Zbl 1024.35070 [38] R. M. Colombo, A. Corli: On $$2\times 2$$ conservation laws with large data. NoDEA Nonlinear Differential Equations Appl. 10 (2003), 255-268. · Zbl 1024.35071 [39] R. M. Colombo, A. Corli: A semilinear structure on semigroups in a metric space. Semigroup Forum · Zbl 1069.47065 [40] R. M. Colombo, A. Groli: Minimising stop & go waves to optimise traffic flow. Appl. Math. Letters · Zbl 1058.49011 [41] R. M. Colombo, A. Groli: On the initial boundary value problem for Temple systems. Nonlinear Anal. 56 (2004), 567-589. · Zbl 1058.35150 [42] R. M. Colombo, A. Groli: On the optimization of a conservation law. Calc. Var. Partial Differential Equations 19 (2004), 269-280. · Zbl 1352.35079 [43] R. M. Colombo, F. S. Priuli: Characterization of Riemann solvers for the two phase $$p$$-system. Comm. Partial Differential Equations 28 (2003), 1371-1389. · Zbl 1030.35117 [44] R. M. Colombo, N. H. Risebro: Continuous dependence in the large for some equations of gas dynamics. Comm. Partial Differential Equations 23 (1998), 1693-1718. · Zbl 0927.35082 [45] G. Crasta, B. Piccoli: Viscosity solutions and uniqueness for systems of inhomogeneous balance laws. Discrete Contin. Dynam. Systems 3 (1997), 477-502. · Zbl 0949.35089 [46] C. M. Dafermos: Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38 (1972), 33-41. · Zbl 0233.35014 [47] C. M. Dafermos: Generalized characteristics in hyperbolic systems of conservation laws. Arch. Rational Mech. Anal. 107 (1989), 127-155. · Zbl 0714.35046 [48] C. M. Dafermos: Hyperbolic Conservation Laws in Continuum Physics. Springer-Verlag, Berlin, first edition, 2000. · Zbl 0940.35002 [49] C. M. Dafermos, X. Geng: Generalised characteristics in hyperbolic systems of conservation laws with special coupling. Proc. Roy. Soc. Edinburgh Sect. A 116 (1990), 245-278. · Zbl 0726.35080 [50] C. M. Dafermos, X. Geng: Generalized characteristics, uniqueness and regularity of solutions in a hyperbolic system of conservation laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 231-269. · Zbl 0776.35033 [51] F. Dubois, P. LeFloch: Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71 (1988), 93-122. · Zbl 0649.35057 [52] J. Glimm: Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pure Appl. Math. 18 (1965), 697-715. · Zbl 0141.28902 [53] H. Holden, N. H. Risebro: Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences Vol. 152. Springer-Verlag, Berlin, 2002. [54] H. K. Jenssen: Blowup for systems of conservation laws. SIAM J. Math. Anal. 31 (2000), 894-908 · Zbl 0969.35091 [55] P. D. Lax: Hyperbolic systems of conservation laws II. Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803 [56] P. G. LeFloch: Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical shock waves Lectures in Mathematics ETH Zürich. Birkhäuser-Verlag, Basel, 2002. · Zbl 1019.35001 [57] M. Lewicka: $$L^1$$ stability of patterns of non-interacting large shock waves. Indiana Univ. Math. J. 49 (2000), 1515-1537. · Zbl 0987.35108 [58] M. Lewicka, K. Trivisa: On the $$L^1$$ well-posedness of systems of conservation laws near solutions containing two large shocks. J. Differential Equations 179 (2002), 133-177. · Zbl 0994.35086 [59] T.-P. Liu: The deterministic version of the Glimm scheme. Comm. Math. Phys. 57 (1977), 135-148. · Zbl 0376.35042 [60] T.-P. Liu, T. Yang: $$L^1$$ stability for $$2\times 2$$ systems of hyperbolic conservation laws. J. Amer. Math. Soc. 12 (1999), 729-774. · Zbl 0924.35083 [61] T.-P. Liu, T. Yang: A new entropy functional for a scalar conservation law. Comm. Pure Appl. Math. 52 (1999), 1427-1442. DOI 10.1002/(SICI)1097-0312(199911)52:113.0.CO;2-R | [62] T.-P. Liu, T. Yang: Well-posedness theory for hyperbolic conservation laws. Comm. Pure Appl. Math. 52 (1999), 1553-1586. DOI 10.1002/(SICI)1097-0312(199912)52:123.0.CO;2-S | [63] Y. Lu: Hyperbolic Conservation Laws and the Compensated Compactness Method. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 128. Chapman & Hall/CRC, Boca Raton, 2003. · Zbl 1024.35002 [64] N. H. Risebro: A front-tracking alternative to the random choice method. Proc. Amer. Math. Soc. 117 (1993), 1125-1139. · Zbl 0799.35153 [65] S. Schochet: The essence of Glimm’s scheme. Nonlinear Evolutionary Partial Differential Equations (Beijing, 1993), AMS/IP Stud. Adv. Math., Vol. 3, Amer. Math. Soc., Providence, 1997, pp. 355-362. · Zbl 0882.35009 [66] D. Serre: Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation. J. Differential Equations 68 (1987), 137-168. · Zbl 0627.35062 [67] D. Serre: Systems of Conservation Laws, 1 & 2. Cambridge University Press, Cambridge, 1999, 2000, · Zbl 1063.35520 [68] M. Slemrod: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 81 (1983), 301-315. · Zbl 0505.76082 [69] B. Temple: Systems of conservation laws with invariant submanifolds. Trans. Amer. Math. Soc. 280 (1983), 781-795. · Zbl 0559.35046 [70] B. Temple: $$L^1$$-contractive metrics for systems of conservation laws. Trans. Amer. Math. Soc. 288 (1985), 471-480. · Zbl 0568.35065 [71] Z. H. Teng, A. J. Chorin, and T. P. Liu: Riemann problems for reacting gas, with applications to transition. SIAM J. Appl. Math. 42 (1982), 964-981. · Zbl 0502.76088 [72] K. Trivisa: A priori estimates in hyperbolic systems of conservation laws via generalized characteristics. Comm. Partial Differential Equations 22 (1997), 235-267. · Zbl 0881.35018 [73] L. Truskinovsky: About the “normal growth” approximation in the dynamical theory of phase transitions. Contin. Mech. Thermodyn. 6 (1994), 185-208. · Zbl 0877.73006
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