The semigroup generated by \(2 \times 2\) conservation laws. (English) Zbl 0849.35068
The Cauchy problem for a strictly hyperbolic \(2\times 2\) system of conservation laws in one space dimension is considered, assuming that each characteristic field is either linearly degenerate or genuinely nonlinear. A new algorithm is given, based on the wavefront tracking, which yields that the Cauchy sequence of approximate solutions converge to a unique limit, depending continuously on the initial data. The solutions constitute a continuous semigroup, defined on a domain \(D\subset L^1(\mathbb{R}, \mathbb{R}^2)\).
Reviewer: S.Tersian (Russe)
MSC:
| 35L50 | Initial-boundary value problems for first-order hyperbolic systems |
| 35L65 | Hyperbolic conservation laws |
References:
| [1] | A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988), 409-421. · Zbl 0632.35041 · doi:10.1512/iumj.1988.37.37021 |
| [2] | A. Bressan, Global solutions of systems of conservation laws by wave-front tracking, J. Math. Anal. Appl. 170 (1992), 414-432. · Zbl 0779.35067 · doi:10.1016/0022-247X(92)90027-B |
| [3] | A. Bressan, A contractive metric for systems of conservation laws with coinciding shock and rarefaction waves, J. Diff. Eqs. 106 (1993), 332-366. · Zbl 0802.35095 · doi:10.1006/jdeq.1993.1111 |
| [4] | A. Bressan, A locally contractive metric for systems of conservation laws, Ann. Scuola Norm. Sup. Pisa, Serie IV, 22 (1995), 109-135. · Zbl 0867.35060 |
| [5] | A. Bressan, The unique limit of the Glimm scheme, Arch. Rational Mech. Anal. 130 (1995), 205-230. · Zbl 0835.35088 · doi:10.1007/BF00392027 |
| [6] | M. Crandall, The semigroup approach to first-order quasilinear equations in several space variables, Israel J. Math. 12 (1972), 108-132. · Zbl 0246.35018 · doi:10.1007/BF02764657 |
| [7] | C. 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 · doi:10.1016/0022-247X(72)90114-X |
| [8] | R. DiPerna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 60 (1975), 75-100. · Zbl 0324.35062 · doi:10.1007/BF00281470 |
| [9] | R. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Diff. Eqs. 20 (1976), 187-212. · Zbl 0314.58010 · doi:10.1016/0022-0396(76)90102-9 |
| [10] | R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal 82 (1983), 27-70. · Zbl 0519.35054 · doi:10.1007/BF00251724 |
| [11] | J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697-715. · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 |
| [12] | J. Glimm & P. Lax, Decay of solutions of systems of hyperbolic conservation laws, Memoirs Amer. Math. Soc. 101, 1970. · Zbl 0204.11304 |
| [13] | S. Kru?kov, First order quasilinear equations with several space variables, Math. USSR Sbornik 10 (1970), 217-243. · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156 |
| [14] | P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566. · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 |
| [15] | T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135-148. · Zbl 0376.35042 · doi:10.1007/BF01625772 |
| [16] | T.-P. Liu, Decay to N-waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure. Appl. Math. 30 (1977), 585-610. · Zbl 0357.35059 · doi:10.1002/cpa.3160300505 |
| [17] | T.-P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 767-796. · Zbl 0358.35014 · doi:10.1002/cpa.3160300605 |
| [18] | V. J. Ljapidevskii, On correctness classes for nonlinear hyperbolic systems, Soviet Math. Dokl. 16 (1975), 1505-1509. · Zbl 0329.35042 |
| [19] | F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa 5 (1978), 489-507. · Zbl 0399.46022 |
| [20] | G. Pimbley, A semigroup for Lagrangian 1D isentropic flow, in Transport theory, invariant imbedding and integral equations, G. Webb ed., Dekker, New York, 1989. · Zbl 0689.76003 |
| [21] | N. H. Risebro, A front-tracking alternative to the random choice method, Proc. Amer. Math, Soc. 117 (1993), 1125-1139. · Zbl 0799.35153 · doi:10.1090/S0002-9939-1993-1120511-X |
| [22] | B. L. Ro?destvenskii & N. Yanenko, Systems of quasilinear equations, Amer. Math. Soc., 1983. |
| [23] | J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983. · Zbl 0508.35002 |
| [24] | L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 4, R. J. Knops, ed., Pitman Press (1979). · Zbl 0437.35004 |
| [25] | B. Temple, No L 1-contractive metrics for systems of conservation laws, Trans. Amer. Math. Soc. 288 (1985), 471-480. · Zbl 0568.35065 |
| [26] | E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. I, Springer-Verlag, 1993. · Zbl 0794.47033 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
