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

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.
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. Google Scholar

[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. Google Scholar

[3]

A. Bressan, Contractive metrics for nonlinear hyperbolic systems,, Indiana Univ. Math. J., 37 (1988), 409. doi: 10.1512/iumj.1988.37.37021. Google Scholar

[4]

A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar

[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). Google Scholar

[6]

S. Chaplygin, On gas jets,, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

D. X. Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems,", MSJ Memoirs, (2000). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

T. T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", Research in Applied Mathematics, (1994). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems,", Duke University Mathematics Series V, (1985). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

D. Serre, "Systems of Conservation Laws $I, II$,", Cambridge University Press, (2000). Google Scholar

[29]

D. Serre, Multidimensional shock interaction for a Chaplygin gas,, Arch. Rational Mech. Anal., 191 (2009), 539. doi: 10.1007/s00205-008-0110-z. Google Scholar

[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. Google Scholar

[31]

H. S. Tsien, Two dimensional subsonic flow of compressible fluids,, J. Aeron. Sci., 6 (1939), 399. Google Scholar

[32]

T. von Karman, Compressibility effects in aerodynamics,, J. Aeron. Sci., 8 (1941), 337. Google Scholar

[33]

Y. Zhou, The Goursat problem for reducible quasilinear hyperbolic systems,, Chin. Ann. Math. Ser. A, 13 (1992), 437. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

A. Bressan, Contractive metrics for nonlinear hyperbolic systems,, Indiana Univ. Math. J., 37 (1988), 409. doi: 10.1512/iumj.1988.37.37021. Google Scholar

[4]

A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, (2000). Google Scholar

[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). Google Scholar

[6]

S. Chaplygin, On gas jets,, Sci. Mem. Moscow Univ. Math. Phys., 21 (1904), 1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

D. X. Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems,", MSJ Memoirs, (2000). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

T. T. Li, "Global Classical Solutions for Quasilinear Hyperbolic Systems,", Research in Applied Mathematics, (1994). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems,", Duke University Mathematics Series V, (1985). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[28]

D. Serre, "Systems of Conservation Laws $I, II$,", Cambridge University Press, (2000). Google Scholar

[29]

D. Serre, Multidimensional shock interaction for a Chaplygin gas,, Arch. Rational Mech. Anal., 191 (2009), 539. doi: 10.1007/s00205-008-0110-z. Google Scholar

[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. Google Scholar

[31]

H. S. Tsien, Two dimensional subsonic flow of compressible fluids,, J. Aeron. Sci., 6 (1939), 399. Google Scholar

[32]

T. von Karman, Compressibility effects in aerodynamics,, J. Aeron. Sci., 8 (1941), 337. Google Scholar

[33]

Y. Zhou, The Goursat problem for reducible quasilinear hyperbolic systems,, Chin. Ann. Math. Ser. A, 13 (1992), 437. Google Scholar

[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. Google Scholar

[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. Google Scholar

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