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Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data
1. | Department of Mathematics, Fuzhou University, Fuzhou 350002, China |
References:
[1] |
D. Amadori and W. Shen, The slow erosion limit in a model of granular flow, Arch. Ration. Mech. Anal., 199 (2011), 1-31.
doi: 10.1007/s00205-010-0313-y. |
[2] |
D. Amadori and W. Shen, Global existence of large BV solutions in a model of granular flow, Comm. Partial Differential Equations, 34 (2009), 1003-1040.
doi: 10.1080/03605300902892279. |
[3] |
A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J., 37 (1988), 409-421.
doi: 10.1512/iumj.1988.37.37021. |
[4] |
A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. |
[5] |
T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics," Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, New York: Longman Scientific and Technical, 1989. |
[6] |
S. Chaplygin, On gas jets, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1-121. |
[7] |
G. Q. Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98.
doi: 10.1007/s00205-002-0229-2. |
[8] |
W. R. Dai, Asymptotic behavior of global classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chin. Ann. Math. Ser. B., 27 (2006), 263-286.
doi: 10.1007/s11401-004-0523-4. |
[9] |
W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, J. Differential Equations, 235 (2007), 127-165.
doi: 10.1016/j.jde.2006.12.020. |
[10] |
Y. Z. Duan, Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries, J. Math. Anal. Appl., 351 (2009), 186-205.
doi: 10.1016/j.jmaa.2008.10.012. |
[11] |
Y. Z. Duan and K. C. Xu, Long time behavior of classical solutions to the generalized Goursat problem of quasilinear hyperbolic systems, Nonlinear Anal., 72 (2010), 209-230.
doi: 10.1016/j.na.2009.06.048. |
[12] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.
doi: 10.1002/cpa.3160180408. |
[13] |
J. Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Bull. Amer. Math. Soc., 73 (1967), 105.
doi: 10.1090/S0002-9904-1967-11666-5. |
[14] |
D. X. Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems," MSJ Memoirs, vol. 6, Mathematical Society of Japan, Tokyo, 2000. |
[15] |
D. X. Kong, K. F. Liu and Y. Z. Wang, Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases, Sci. China Math., 53 (2010), 719-738.
doi: 10.1007/s11425-010-0060-4. |
[16] |
D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $R^{2+n}$, J. Math. Phys., 47 (2006), 013503.
doi: 10.1063/1.2158435. |
[17] |
D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Partial Differential Equations, 28 (2003), 1203-1220.
doi: 10.1081/PDE-120021192. |
[18] |
T. T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," Research in Applied Mathematics, vol. 32, Wiley/Masson, New York, 1994. |
[19] |
T. T. Li and Y. J. Peng, Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form, Nonlinear Anal., 55 (2003), 937-949.
doi: 10.1016/j.na.2003.08.010. |
[20] |
T. T. Li and Y. J. Peng, The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal., 52 (2003), 573-583.
doi: 10.1016/S0362-546X(02)00123-2. |
[21] |
T. T. Li, Y. J. Peng and J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91 (2009), 553-568.
doi: 10.1016/j.matpur.2009.01.008. |
[22] |
T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential Equations, 19 (1994), 1263-1317.
doi: 10.1080/03605309408821055. |
[23] |
T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal., 28 (1997), 1299-1332.
doi: 10.1016/0362-546X(95)00228-N. |
[24] |
T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems," Duke University Mathematics Series V, Duke University, Durham, 1985. |
[25] |
J. Liu and K. Pan, Global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system, Boundary Value Problems, 2012 (2012), 36.
doi: 10.1186/1687-2770-2012-36. |
[26] |
J. Liu and Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500.
doi: 10.1002/mma.797. |
[27] |
T. Nishida and J. Smoller, Solutions in the large for some nonlinear conservation laws, Comm. Pure Appl. Math., 26 (1973), 183-200.
doi: 10.1002/cpa.3160260205. |
[28] |
D. Serre, "Systems of Conservation Laws $I, II$," Cambridge University Press, Cambridge, 2000. |
[29] |
D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rational Mech. Anal., 191 (2009), 539-577.
doi: 10.1007/s00205-008-0110-z. |
[30] |
Z. Q. Shao, A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Nonlinear Anal., 73 (2010), 600-613.
doi: 10.1016/j.na.2010.03.029. |
[31] |
H. S. Tsien, Two dimensional subsonic flow of compressible fluids, J. Aeron. Sci., 6 (1939), 399-407. |
[32] |
T. von Karman, Compressibility effects in aerodynamics, J. Aeron. Sci., 8 (1941), 337-356. |
[33] |
Y. Zhou, The Goursat problem for reducible quasilinear hyperbolic systems, Chin. Ann. Math. Ser. A, 13 (1992), 437-441. |
[34] |
Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56.
doi: 10.1142/S0252959904000044. |
[35] |
Z. Q. Shao, Asymptotic behaviour of global classical solutions to the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems, IMA J. Appl. Math., 78 (2013), 1-31.
doi: 10.1093/imamat/hxr032. |
show all references
References:
[1] |
D. Amadori and W. Shen, The slow erosion limit in a model of granular flow, Arch. Ration. Mech. Anal., 199 (2011), 1-31.
doi: 10.1007/s00205-010-0313-y. |
[2] |
D. Amadori and W. Shen, Global existence of large BV solutions in a model of granular flow, Comm. Partial Differential Equations, 34 (2009), 1003-1040.
doi: 10.1080/03605300902892279. |
[3] |
A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J., 37 (1988), 409-421.
doi: 10.1512/iumj.1988.37.37021. |
[4] |
A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, 20. Oxford University Press, Oxford, 2000. |
[5] |
T. Chang and L. Hsiao, "The Riemann Problem and Interaction of Waves in Gas Dynamics," Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 41, New York: Longman Scientific and Technical, 1989. |
[6] |
S. Chaplygin, On gas jets, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1-121. |
[7] |
G. Q. Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98.
doi: 10.1007/s00205-002-0229-2. |
[8] |
W. R. Dai, Asymptotic behavior of global classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chin. Ann. Math. Ser. B., 27 (2006), 263-286.
doi: 10.1007/s11401-004-0523-4. |
[9] |
W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, J. Differential Equations, 235 (2007), 127-165.
doi: 10.1016/j.jde.2006.12.020. |
[10] |
Y. Z. Duan, Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries, J. Math. Anal. Appl., 351 (2009), 186-205.
doi: 10.1016/j.jmaa.2008.10.012. |
[11] |
Y. Z. Duan and K. C. Xu, Long time behavior of classical solutions to the generalized Goursat problem of quasilinear hyperbolic systems, Nonlinear Anal., 72 (2010), 209-230.
doi: 10.1016/j.na.2009.06.048. |
[12] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715.
doi: 10.1002/cpa.3160180408. |
[13] |
J. Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Bull. Amer. Math. Soc., 73 (1967), 105.
doi: 10.1090/S0002-9904-1967-11666-5. |
[14] |
D. X. Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems," MSJ Memoirs, vol. 6, Mathematical Society of Japan, Tokyo, 2000. |
[15] |
D. X. Kong, K. F. Liu and Y. Z. Wang, Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases, Sci. China Math., 53 (2010), 719-738.
doi: 10.1007/s11425-010-0060-4. |
[16] |
D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $R^{2+n}$, J. Math. Phys., 47 (2006), 013503.
doi: 10.1063/1.2158435. |
[17] |
D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Partial Differential Equations, 28 (2003), 1203-1220.
doi: 10.1081/PDE-120021192. |
[18] |
T. T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems," Research in Applied Mathematics, vol. 32, Wiley/Masson, New York, 1994. |
[19] |
T. T. Li and Y. J. Peng, Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form, Nonlinear Anal., 55 (2003), 937-949.
doi: 10.1016/j.na.2003.08.010. |
[20] |
T. T. Li and Y. J. Peng, The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal., 52 (2003), 573-583.
doi: 10.1016/S0362-546X(02)00123-2. |
[21] |
T. T. Li, Y. J. Peng and J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl., 91 (2009), 553-568.
doi: 10.1016/j.matpur.2009.01.008. |
[22] |
T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential Equations, 19 (1994), 1263-1317.
doi: 10.1080/03605309408821055. |
[23] |
T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal., 28 (1997), 1299-1332.
doi: 10.1016/0362-546X(95)00228-N. |
[24] |
T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems," Duke University Mathematics Series V, Duke University, Durham, 1985. |
[25] |
J. Liu and K. Pan, Global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system, Boundary Value Problems, 2012 (2012), 36.
doi: 10.1186/1687-2770-2012-36. |
[26] |
J. Liu and Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500.
doi: 10.1002/mma.797. |
[27] |
T. Nishida and J. Smoller, Solutions in the large for some nonlinear conservation laws, Comm. Pure Appl. Math., 26 (1973), 183-200.
doi: 10.1002/cpa.3160260205. |
[28] |
D. Serre, "Systems of Conservation Laws $I, II$," Cambridge University Press, Cambridge, 2000. |
[29] |
D. Serre, Multidimensional shock interaction for a Chaplygin gas, Arch. Rational Mech. Anal., 191 (2009), 539-577.
doi: 10.1007/s00205-008-0110-z. |
[30] |
Z. Q. Shao, A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Nonlinear Anal., 73 (2010), 600-613.
doi: 10.1016/j.na.2010.03.029. |
[31] |
H. S. Tsien, Two dimensional subsonic flow of compressible fluids, J. Aeron. Sci., 6 (1939), 399-407. |
[32] |
T. von Karman, Compressibility effects in aerodynamics, J. Aeron. Sci., 8 (1941), 337-356. |
[33] |
Y. Zhou, The Goursat problem for reducible quasilinear hyperbolic systems, Chin. Ann. Math. Ser. A, 13 (1992), 437-441. |
[34] |
Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56.
doi: 10.1142/S0252959904000044. |
[35] |
Z. Q. Shao, Asymptotic behaviour of global classical solutions to the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems, IMA J. Appl. Math., 78 (2013), 1-31.
doi: 10.1093/imamat/hxr032. |
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