# American Institute of Mathematical Sciences

February  2016, 9(1): 15-32. doi: 10.3934/dcdss.2016.9.15

## The path decomposition technique for systems of hyperbolic conservation laws

 1 Department of Engineering Science, Osaka Electro-Communication University, 18-8 Hatucho, Neyagawa, Osaka 572-8530, Japan 2 Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy

Received  September 2014 Revised  February 2015 Published  December 2015

We are concerned with the problem of the global (in time) existence of weak solutions to hyperbolic systems of conservation laws, in one spatial dimension. First, we provide a survey of the different facets of a technique that has been used in several papers in the last years: the path decomposition. Then, we report on two very recent results that have been achieved by means of suitable applications of this technique. The first one concerns a system of three equations arising in the dynamic modeling of phase transitions, the second one is the famous Euler system for nonisentropic fluid flow. In both cases, the results concern classes of initial data with possibly large total variation.
Citation: Fumioki Asakura, Andrea Corli. The path decomposition technique for systems of hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 15-32. doi: 10.3934/dcdss.2016.9.15
##### References:
 [1] D. Amadori and A. Corli, On a model of multiphase flow,, SIAM J. Math. Anal., 40 (2008), 134. doi: 10.1137/07069211X. [2] D. Amadori and A. Corli, Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow,, Nonlinear Anal., 72 (2010), 2527. doi: 10.1016/j.na.2009.10.048. [3] D. Amadori and G. Guerra, Global BV solutions and relaxation limit for a system of conservation laws,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1. doi: 10.1017/S0308210500000767. [4] F. Asakura, Decay of solutions for the equations of isothermal gas dynamics,, Japan J. Indust. Appl. Math., 10 (1993), 133. doi: 10.1007/BF03167207. [5] F. Asakura, Wave-front tracking for the equations of isentropic gas dynamics,, Quart. Appl. Math., 63 (2005), 20. doi: 10.1090/S0033-569X-04-00935-8. [6] F. Asakura, Wave-front tracking for the equations of non-isentropic gas dynamics-basic lemmas,, Acta Math. Vietnam., 38 (2013), 487. doi: 10.1007/s40306-013-0030-3. [7] F. Asakura and A. Corli, Global existence of solutions by path decomposition for a model of multiphase flow,, Quart. Appl. Math., 71 (2013), 135. doi: 10.1090/S0033-569X-2012-01318-4. [8] F. Asakura and A. Corli, Wave-front tracking for the equations of non-isentropic gas dynamics,, Ann. Mat. Pura Appl., 194 (2015), 581. doi: 10.1007/s10231-013-0390-2. [9] A. Bressan, Global solutions of systems of conservation laws by wave-front tracking,, J. Math. Anal. Appl., 170 (1992), 414. doi: 10.1016/0022-247X(92)90027-B. [10] A. Bressan, The unique limit of the Glimm scheme,, Arch. Rational Mech. Anal., 130 (1995), 205. doi: 10.1007/BF00392027. [11] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford University Press, (2000). [12] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Third edition, (2010). doi: 10.1007/978-3-642-04048-1. [13] R. J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws,, J. Differential Equations, 20 (1976), 187. doi: 10.1016/0022-0396(76)90102-9. [14] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697. doi: 10.1002/cpa.3160180408. [15] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,, Mat. Sb. (N.S.), 47 (1959), 271. [16] P. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., (1971), 603. [17] P. D. Lax, Hyperbolic systems of conservation laws. II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. [18] T. P. Liu, The deterministic version of the Glimm scheme,, Comm. Math. Phys., 57 (1977), 135. doi: 10.1007/BF01625772. [19] T. P. Liu, Initial-boundary value problems for gas dynamics,, Arch. Rational Mech. Anal., 64 (1977), 137. doi: 10.1007/BF00280095. [20] T. P. Liu, Solutions in the large for the equations of nonisentropic gas dynamics,, Indiana Univ. Math. J., 26 (1977), 147. doi: 10.1512/iumj.1977.26.26011. [21] T. Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system,, Proc. Japan Acad., 44 (1968), 642. doi: 10.3792/pja/1195521083. [22] T. Nishida and J. A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws,, Comm. Pure Appl. Math., 26 (1973), 183. doi: 10.1002/cpa.3160260205. [23] N. H. Risebro, A front-tracking alternative to the random choice method,, Proc. Amer. Math. Soc., 117 (1993), 1125. doi: 10.1090/S0002-9939-1993-1120511-X. [24] J. B. Temple, Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics,, J. Differential Equations, 41 (1981), 96. doi: 10.1016/0022-0396(81)90055-3. [25] B. Temple and R. Young, The large time stability of sound waves,, Comm. Math. Phys., 179 (1996), 417. doi: 10.1007/BF02102596. [26] R. Young, Sup-norm stability for Glimm's scheme,, Comm. Pure Appl. Math., 46 (1993), 903. doi: 10.1002/cpa.3160460605.

show all references

##### References:
 [1] D. Amadori and A. Corli, On a model of multiphase flow,, SIAM J. Math. Anal., 40 (2008), 134. doi: 10.1137/07069211X. [2] D. Amadori and A. Corli, Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow,, Nonlinear Anal., 72 (2010), 2527. doi: 10.1016/j.na.2009.10.048. [3] D. Amadori and G. Guerra, Global BV solutions and relaxation limit for a system of conservation laws,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1. doi: 10.1017/S0308210500000767. [4] F. Asakura, Decay of solutions for the equations of isothermal gas dynamics,, Japan J. Indust. Appl. Math., 10 (1993), 133. doi: 10.1007/BF03167207. [5] F. Asakura, Wave-front tracking for the equations of isentropic gas dynamics,, Quart. Appl. Math., 63 (2005), 20. doi: 10.1090/S0033-569X-04-00935-8. [6] F. Asakura, Wave-front tracking for the equations of non-isentropic gas dynamics-basic lemmas,, Acta Math. Vietnam., 38 (2013), 487. doi: 10.1007/s40306-013-0030-3. [7] F. Asakura and A. Corli, Global existence of solutions by path decomposition for a model of multiphase flow,, Quart. Appl. Math., 71 (2013), 135. doi: 10.1090/S0033-569X-2012-01318-4. [8] F. Asakura and A. Corli, Wave-front tracking for the equations of non-isentropic gas dynamics,, Ann. Mat. Pura Appl., 194 (2015), 581. doi: 10.1007/s10231-013-0390-2. [9] A. Bressan, Global solutions of systems of conservation laws by wave-front tracking,, J. Math. Anal. Appl., 170 (1992), 414. doi: 10.1016/0022-247X(92)90027-B. [10] A. Bressan, The unique limit of the Glimm scheme,, Arch. Rational Mech. Anal., 130 (1995), 205. doi: 10.1007/BF00392027. [11] A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford University Press, (2000). [12] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Third edition, (2010). doi: 10.1007/978-3-642-04048-1. [13] R. J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws,, J. Differential Equations, 20 (1976), 187. doi: 10.1016/0022-0396(76)90102-9. [14] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697. doi: 10.1002/cpa.3160180408. [15] S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics,, Mat. Sb. (N.S.), 47 (1959), 271. [16] P. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., (1971), 603. [17] P. D. Lax, Hyperbolic systems of conservation laws. II,, Comm. Pure Appl. Math., 10 (1957), 537. doi: 10.1002/cpa.3160100406. [18] T. P. Liu, The deterministic version of the Glimm scheme,, Comm. Math. Phys., 57 (1977), 135. doi: 10.1007/BF01625772. [19] T. P. Liu, Initial-boundary value problems for gas dynamics,, Arch. Rational Mech. Anal., 64 (1977), 137. doi: 10.1007/BF00280095. [20] T. P. Liu, Solutions in the large for the equations of nonisentropic gas dynamics,, Indiana Univ. Math. J., 26 (1977), 147. doi: 10.1512/iumj.1977.26.26011. [21] T. Nishida, Global solution for an initial boundary value problem of a quasilinear hyperbolic system,, Proc. Japan Acad., 44 (1968), 642. doi: 10.3792/pja/1195521083. [22] T. Nishida and J. A. Smoller, Solutions in the large for some nonlinear hyperbolic conservation laws,, Comm. Pure Appl. Math., 26 (1973), 183. doi: 10.1002/cpa.3160260205. [23] N. H. Risebro, A front-tracking alternative to the random choice method,, Proc. Amer. Math. Soc., 117 (1993), 1125. doi: 10.1090/S0002-9939-1993-1120511-X. [24] J. B. Temple, Solutions in the large for the nonlinear hyperbolic conservation laws of gas dynamics,, J. Differential Equations, 41 (1981), 96. doi: 10.1016/0022-0396(81)90055-3. [25] B. Temple and R. Young, The large time stability of sound waves,, Comm. Math. Phys., 179 (1996), 417. doi: 10.1007/BF02102596. [26] R. Young, Sup-norm stability for Glimm's scheme,, Comm. Pure Appl. Math., 46 (1993), 903. doi: 10.1002/cpa.3160460605.
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