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

Zhi-Qiang Shao ^1,

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 and 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), 1-31. 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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