Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data

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November 2013, 12(6): 2739-2752. doi: 10.3934/cpaa.2013.12.2739

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

Received October 2012 Revised December 2012 Published May 2013

Abstract
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We investigate the existence of a global classical solution to the Goursat problem for linearly degenerate quasilinear hyperbolic systems of diagonal form. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409-421] suggests that one may achieve global smoothness even if the $C^1$ norm of the initial data is large, we prove that, if the $C^1$ norm and the BV norm of the boundary data are bounded but possibly large, then the solution remains $C^1$ globally in time. Applications include the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ and the one-dimensional Chaplygin gas equations.

Keywords: large BV data, Chaplygin gas equations., Quasilinear hyperbolic system of diagonal form, Goursat problem, global classical solution, linear degeneracy.

Mathematics Subject Classification: Primary: 35L45, 35L60; Secondary: 35Q4.

Citation: Zhi-Qiang Shao. Global existence of classical solutions of Goursat problem for quasilinear hyperbolic systems of diagonal form with large BV data. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2739-2752. doi: 10.3934/cpaa.2013.12.2739

References:

[1]	D. Amadori and W. Shen, The slow erosion limit in a model of granular flow,, Arch. Ration. Mech. Anal., 199 (2011), 1. 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. doi: 10.1080/03605300902892279.
[3]	A. Bressan, Contractive metrics for nonlinear hyperbolic systems,, Indiana Univ. Math. J., 37 (1988), 409. 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, (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, (1989).
[6]	S. Chaplygin, On gas jets,, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1.
[7]	G. Q. Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations,, Arch. Rational Mech. Anal., 166 (2003), 81. 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. 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. 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. 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. 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. 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). doi: 10.1090/S0002-9904-1967-11666-5.
[14]	D. X. Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems,", MSJ Memoirs, (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. 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). 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. doi: 10.1081/PDE-120021192.
[18]	T. T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", Research in Applied Mathematics, (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. 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. 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. 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. 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. 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, (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). 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. 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. doi: 10.1002/cpa.3160260205.
[28]	D. Serre, "Systems of Conservation Laws $I, II$,", Cambridge University Press, (2000).
[29]	D. Serre, Multidimensional shock interaction for a Chaplygin gas,, Arch. Rational Mech. Anal., 191 (2009), 539. 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. 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.
[32]	T. von Karman, Compressibility effects in aerodynamics,, J. Aeron. Sci., 8 (1941), 337.
[33]	Y. Zhou, The Goursat problem for reducible quasilinear hyperbolic systems,, Chin. Ann. Math. Ser. A, 13 (1992), 437.
[34]	Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chinese Ann. Math. Ser. B, 25 (2004), 37. 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. 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. 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. doi: 10.1080/03605300902892279.
[3]	A. Bressan, Contractive metrics for nonlinear hyperbolic systems,, Indiana Univ. Math. J., 37 (1988), 409. 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, (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, (1989).
[6]	S. Chaplygin, On gas jets,, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1.
[7]	G. Q. Chen and P. G. LeFloch, Existence theory for the isentropic Euler equations,, Arch. Rational Mech. Anal., 166 (2003), 81. 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. 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. 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. 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. 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. 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). doi: 10.1090/S0002-9904-1967-11666-5.
[14]	D. X. Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems,", MSJ Memoirs, (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. 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). 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. doi: 10.1081/PDE-120021192.
[18]	T. T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", Research in Applied Mathematics, (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. 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. 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. 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. 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. 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, (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). 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. 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. doi: 10.1002/cpa.3160260205.
[28]	D. Serre, "Systems of Conservation Laws $I, II$,", Cambridge University Press, (2000).
[29]	D. Serre, Multidimensional shock interaction for a Chaplygin gas,, Arch. Rational Mech. Anal., 191 (2009), 539. 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. 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.
[32]	T. von Karman, Compressibility effects in aerodynamics,, J. Aeron. Sci., 8 (1941), 337.
[33]	Y. Zhou, The Goursat problem for reducible quasilinear hyperbolic systems,, Chin. Ann. Math. Ser. A, 13 (1992), 437.
[34]	Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chinese Ann. Math. Ser. B, 25 (2004), 37. 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. doi: 10.1093/imamat/hxr032.

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