# American Institute of Mathematical Sciences

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
 [1] Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001 [2] Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026 [3] Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2733-2748. doi: 10.3934/cpaa.2014.13.2733 [4] Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 [5] Marko Nedeljkov, Sanja Ružičić. On the uniqueness of solution to generalized Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4439-4460. doi: 10.3934/dcds.2017190 [6] Lihui Guo, Wancheng Sheng, Tong Zhang. The two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system$^*$. Communications on Pure & Applied Analysis, 2010, 9 (2) : 431-458. doi: 10.3934/cpaa.2010.9.431 [7] Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59 [8] Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016 [9] Dan-Andrei Geba, Manoussos G. Grillakis. Large data global regularity for the classical equivariant Skyrme model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5537-5576. doi: 10.3934/dcds.2018244 [10] Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945 [11] Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501 [12] Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127 [13] Lihui Guo, Tong Li, Gan Yin. The vanishing pressure limits of Riemann solutions to the Chaplygin gas equations with a source term. Communications on Pure & Applied Analysis, 2017, 16 (1) : 295-310. doi: 10.3934/cpaa.2017014 [14] Yanbo Hu, Tong Li. The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3317-3336. doi: 10.3934/cpaa.2019149 [15] Gary Lieberman. Nonlocal problems for quasilinear parabolic equations in divergence form. Conference Publications, 2003, 2003 (Special) : 563-570. doi: 10.3934/proc.2003.2003.563 [16] Yuusuke Sugiyama. Degeneracy in finite time of 1D quasilinear wave equations Ⅱ. Evolution Equations & Control Theory, 2017, 6 (4) : 615-628. doi: 10.3934/eect.2017031 [17] Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77 [18] Meixiang Huang, Zhi-Qiang Shao. Riemann problem for the relativistic generalized Chaplygin Euler equations. Communications on Pure & Applied Analysis, 2016, 15 (1) : 127-138. doi: 10.3934/cpaa.2016.15.127 [19] Jin Feng He, Wei Xu, Zhi Guo Feng, Xinsong Yang. On the global optimal solution for linear quadratic problems of switched system. Journal of Industrial & Management Optimization, 2019, 15 (2) : 817-832. doi: 10.3934/jimo.2018072 [20] Qin Wang, Kyungwoo Song. The regularity of sonic curves for the two-dimensional Riemann problems of the nonlinear wave system of Chaplygin gas. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1661-1675. doi: 10.3934/dcds.2016.36.1661

2018 Impact Factor: 0.925