American Institute of Mathematical Sciences

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

The path decomposition technique for systems of hyperbolic conservation laws

Fumioki Asakura 1, and  Andrea Corli 2,

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.
Keywords: weak solutions, Hyperbolic systems of conservation laws, global existence..
Mathematics Subject Classification: Primary: 35L65, 35L67; Secondary: 76T30, 35D30, 35Q3.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Bressan, The unique limit of the Glimm scheme,, Arch. Rational Mech. Anal., 130 (1995), 205.  doi: 10.1007/BF00392027.  Google Scholar

[11]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford University Press, (2000).   Google Scholar

[12]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Third edition, (2010).  doi: 10.1007/978-3-642-04048-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

P. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., (1971), 603.   Google Scholar

[17]

P. D. Lax, Hyperbolic systems of conservation laws. II,, Comm. Pure Appl. Math., 10 (1957), 537.  doi: 10.1002/cpa.3160100406.  Google Scholar

[18]

T. P. Liu, The deterministic version of the Glimm scheme,, Comm. Math. Phys., 57 (1977), 135.  doi: 10.1007/BF01625772.  Google Scholar

[19]

T. P. Liu, Initial-boundary value problems for gas dynamics,, Arch. Rational Mech. Anal., 64 (1977), 137.  doi: 10.1007/BF00280095.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. Temple and R. Young, The large time stability of sound waves,, Comm. Math. Phys., 179 (1996), 417.  doi: 10.1007/BF02102596.  Google Scholar

[26]

R. Young, Sup-norm stability for Glimm's scheme,, Comm. Pure Appl. Math., 46 (1993), 903.  doi: 10.1002/cpa.3160460605.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Bressan, The unique limit of the Glimm scheme,, Arch. Rational Mech. Anal., 130 (1995), 205.  doi: 10.1007/BF00392027.  Google Scholar

[11]

A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,, Oxford University Press, (2000).   Google Scholar

[12]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Third edition, (2010).  doi: 10.1007/978-3-642-04048-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

P. Lax, Shock waves and entropy,, in Contributions to Nonlinear Functional Analysis (Proc. Sympos., (1971), 603.   Google Scholar

[17]

P. D. Lax, Hyperbolic systems of conservation laws. II,, Comm. Pure Appl. Math., 10 (1957), 537.  doi: 10.1002/cpa.3160100406.  Google Scholar

[18]

T. P. Liu, The deterministic version of the Glimm scheme,, Comm. Math. Phys., 57 (1977), 135.  doi: 10.1007/BF01625772.  Google Scholar

[19]

T. P. Liu, Initial-boundary value problems for gas dynamics,, Arch. Rational Mech. Anal., 64 (1977), 137.  doi: 10.1007/BF00280095.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. Temple and R. Young, The large time stability of sound waves,, Comm. Math. Phys., 179 (1996), 417.  doi: 10.1007/BF02102596.  Google Scholar

[26]

R. Young, Sup-norm stability for Glimm's scheme,, Comm. Pure Appl. Math., 46 (1993), 903.  doi: 10.1002/cpa.3160460605.  Google Scholar

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