Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm

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October 2012, 32(10): 3501-3524. doi: 10.3934/dcds.2012.32.3501

Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm

Wen-Rong Dai ^1,

1.	Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received May 2011 Revised November 2011 Published May 2012

Abstract
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In this paper, we investigate the formation of singularities of the classical solution to the Cauchy problem of quasi-linear hyperbolic system and give a sharp limit formula for the lifespan of the classical solution. It is important that we only require that the initial data are sufficiently small in the $L^1$ sense and the BV sense.

Keywords: Quasi-linear hyperbolic system, lifespan., normalized coordinates, blow-up, classical solutions, formation of singularities.

Mathematics Subject Classification: Primary: 35L45, 35L60; Secondary: 35L4.

Citation: Wen-Rong Dai. Formation of singularities to quasi-linear hyperbolic systems with initial data of small BV norm. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3501-3524. doi: 10.3934/dcds.2012.32.3501

References:

[1]	S. Bianchini and A. Bressan, Vanishing viscousity solutions to nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-242. doi: 10.4007/annals.2005.161.223.
[2]	A. Bressan, Contractive metrices for nonlinear hyperbolic systems, Indiana U. Math. J., 37 (1988), 409-421. doi: 10.1512/iumj.1988.37.37021.
[3]	A. Bressan, "Hyerbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem," Oxford University Press, 2000.
[4]	Wen-Rong Dai, Asymptotic behavior of classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chinese Ann. Math. B, 27 (2006), 263-286. doi: 10.1007/s11401-004-0523-4.
[5]	Wen-Rong Dai, Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity, J. Math. Anal. Appl., 327 (2007), 188-202. doi: 10.1016/j.jmaa.2006.04.014.
[6]	Wen-Rong Dai, Analysis of singularities for one-dimensional quasilinear hyperbolic systems, J. Math. Anal. Appl., 362 (2010), 72-89. doi: 10.1016/j.jmaa.2009.08.035.
[7]	Wen-Rong Dai and De-Xing Kong, Global existence and asymptotic behavior of classical solutions of quasi-linear hyperbolic systems with linearly degenerate characteristic fields, J. Differ. Equations, 235 (2007), 127-165. doi: 10.1016/j.jde.2006.12.020.
[8]	L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture Notes in Mathematics, 1256 (1987), 214-280. doi: 10.1007/BFb0077745.
[9]	F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Commun. Pur. Appl. Math., 27 (1974), 377-405. doi: 10.1002/cpa.3160270307.
[10]	De-xing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. Math. B, 21 (2000), 413-440. doi: 10.1142/S0252959900000431.
[11]	De-xing Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems," MSJ Memoirs, 6, Mathematical Society of Japan, Tokyo, 2000.
[12]	De-xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems, T. Am. Math. Soc., 354 (2002), 3155-3179. doi: 10.1090/S0002-9947-02-02982-3.
[13]	De-xing Kong and Ta-tsien Li, A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal., 49 (2002), 535-539. doi: 10.1016/S0362-546X(01)00121-3.
[14]	De-xing Kong and Tong Yang, Asymtotic behavior of global classical solutions of quasilinear hyperbolic systems, Commun. Part. Diff. Eq., 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192.
[15]	Ta-Tsien Li and De-xing Kong, Global classical solutions with small amplitude for general quasi-linear hyperbolic systems, in "New Approaches in Nonlinear Analysis," Hardronic Press, (1999), 203-237.
[16]	Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinear nonstrictly hyperbolic systems, Nonlinear Studies, 3 (1996), 203-229.
[17]	Ta-Tsien Li, Yi Zhou and De-Xing Kong, Weak linear degeneracy and global classical solutions for general quasi-linear hyperbolic systems, Commun. Part. Diff. Eq., 19 (1994), 1263-1317. doi: 10.1080/03605309408821055.
[18]	Ta-Tsien Li, Yi Zhou and De-Xing Kong, Global classical solutions for general quasi-linear hyperbolic systems with decay initial data, Nonlinear Anal., 28 (1997), 1299-1332. doi: 10.1016/0362-546X(95)00228-N.
[19]	Tai-Ping Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Differ. Equations, 33 (1979), 92-111. doi: 10.1016/0022-0396(79)90082-2.
[20]	M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana U. Math. J., 34 (1985), 533-589. doi: 10.1512/iumj.1985.34.34030.
[21]	M. Schatzman, The geometry of continuous Glimm functionals, Lectures in Applied Mathematics, 23 (1986), 417-439.
[22]	Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56. doi: 10.1142/S0252959904000469.
[23]	Yi Zhou and Yong-Fu Yang, Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems, Nonlinear Anal., 73 (2010), 1543-1561. doi: 10.1016/j.na.2010.04.057.

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References:

[1]	S. Bianchini and A. Bressan, Vanishing viscousity solutions to nonlinear hyperbolic systems, Ann. Math., 161 (2005), 223-242. doi: 10.4007/annals.2005.161.223.
[2]	A. Bressan, Contractive metrices for nonlinear hyperbolic systems, Indiana U. Math. J., 37 (1988), 409-421. doi: 10.1512/iumj.1988.37.37021.
[3]	A. Bressan, "Hyerbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem," Oxford University Press, 2000.
[4]	Wen-Rong Dai, Asymptotic behavior of classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chinese Ann. Math. B, 27 (2006), 263-286. doi: 10.1007/s11401-004-0523-4.
[5]	Wen-Rong Dai, Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity, J. Math. Anal. Appl., 327 (2007), 188-202. doi: 10.1016/j.jmaa.2006.04.014.
[6]	Wen-Rong Dai, Analysis of singularities for one-dimensional quasilinear hyperbolic systems, J. Math. Anal. Appl., 362 (2010), 72-89. doi: 10.1016/j.jmaa.2009.08.035.
[7]	Wen-Rong Dai and De-Xing Kong, Global existence and asymptotic behavior of classical solutions of quasi-linear hyperbolic systems with linearly degenerate characteristic fields, J. Differ. Equations, 235 (2007), 127-165. doi: 10.1016/j.jde.2006.12.020.
[8]	L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture Notes in Mathematics, 1256 (1987), 214-280. doi: 10.1007/BFb0077745.
[9]	F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Commun. Pur. Appl. Math., 27 (1974), 377-405. doi: 10.1002/cpa.3160270307.
[10]	De-xing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. Math. B, 21 (2000), 413-440. doi: 10.1142/S0252959900000431.
[11]	De-xing Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems," MSJ Memoirs, 6, Mathematical Society of Japan, Tokyo, 2000.
[12]	De-xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems, T. Am. Math. Soc., 354 (2002), 3155-3179. doi: 10.1090/S0002-9947-02-02982-3.
[13]	De-xing Kong and Ta-tsien Li, A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal., 49 (2002), 535-539. doi: 10.1016/S0362-546X(01)00121-3.
[14]	De-xing Kong and Tong Yang, Asymtotic behavior of global classical solutions of quasilinear hyperbolic systems, Commun. Part. Diff. Eq., 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192.
[15]	Ta-Tsien Li and De-xing Kong, Global classical solutions with small amplitude for general quasi-linear hyperbolic systems, in "New Approaches in Nonlinear Analysis," Hardronic Press, (1999), 203-237.
[16]	Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinear nonstrictly hyperbolic systems, Nonlinear Studies, 3 (1996), 203-229.
[17]	Ta-Tsien Li, Yi Zhou and De-Xing Kong, Weak linear degeneracy and global classical solutions for general quasi-linear hyperbolic systems, Commun. Part. Diff. Eq., 19 (1994), 1263-1317. doi: 10.1080/03605309408821055.
[18]	Ta-Tsien Li, Yi Zhou and De-Xing Kong, Global classical solutions for general quasi-linear hyperbolic systems with decay initial data, Nonlinear Anal., 28 (1997), 1299-1332. doi: 10.1016/0362-546X(95)00228-N.
[19]	Tai-Ping Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations, J. Differ. Equations, 33 (1979), 92-111. doi: 10.1016/0022-0396(79)90082-2.
[20]	M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana U. Math. J., 34 (1985), 533-589. doi: 10.1512/iumj.1985.34.34030.
[21]	M. Schatzman, The geometry of continuous Glimm functionals, Lectures in Applied Mathematics, 23 (1986), 417-439.
[22]	Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 25 (2004), 37-56. doi: 10.1142/S0252959904000469.
[23]	Yi Zhou and Yong-Fu Yang, Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems, Nonlinear Anal., 73 (2010), 1543-1561. doi: 10.1016/j.na.2010.04.057.

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