American Institute of Mathematical Sciences

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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[2]

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

[3]

A. Bressan, "Hyerbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem,", Oxford University Press, (2000).   Google Scholar

[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.  doi: 10.1007/s11401-004-0523-4.  Google Scholar

[5]

Wen-Rong Dai, Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity,, J. Math. Anal. Appl., 327 (2007), 188.  doi: 10.1016/j.jmaa.2006.04.014.  Google Scholar

[6]

Wen-Rong Dai, Analysis of singularities for one-dimensional quasilinear hyperbolic systems,, J. Math. Anal. Appl., 362 (2010), 72.  doi: 10.1016/j.jmaa.2009.08.035.  Google Scholar

[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.  doi: 10.1016/j.jde.2006.12.020.  Google Scholar

[8]

L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations,, Lecture Notes in Mathematics, 1256 (1987), 214.  doi: 10.1007/BFb0077745.  Google Scholar

[9]

F. John, Formation of singularities in one-dimensional nonlinear wave propagation,, Commun. Pur. Appl. Math., 27 (1974), 377.  doi: 10.1002/cpa.3160270307.  Google Scholar

[10]

De-xing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data,, Chinese Ann. Math. B, 21 (2000), 413.  doi: 10.1142/S0252959900000431.  Google Scholar

[11]

De-xing Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems,", MSJ Memoirs, 6 (2000).   Google Scholar

[12]

De-xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems,, T. Am. Math. Soc., 354 (2002), 3155.  doi: 10.1090/S0002-9947-02-02982-3.  Google Scholar

[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.  doi: 10.1016/S0362-546X(01)00121-3.  Google Scholar

[14]

De-xing Kong and Tong Yang, Asymtotic behavior of global classical solutions of quasilinear hyperbolic systems,, Commun. Part. Diff. Eq., 28 (2003), 1203.  doi: 10.1081/PDE-120021192.  Google Scholar

[15]

Ta-Tsien Li and De-xing Kong, Global classical solutions with small amplitude for general quasi-linear hyperbolic systems,, in, (1999), 203.   Google Scholar

[16]

Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinear nonstrictly hyperbolic systems,, Nonlinear Studies, 3 (1996), 203.   Google Scholar

[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.  doi: 10.1080/03605309408821055.  Google Scholar

[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.  doi: 10.1016/0362-546X(95)00228-N.  Google Scholar

[19]

Tai-Ping Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations,, J. Differ. Equations, 33 (1979), 92.  doi: 10.1016/0022-0396(79)90082-2.  Google Scholar

[20]

M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem,, Indiana U. Math. J., 34 (1985), 533.  doi: 10.1512/iumj.1985.34.34030.  Google Scholar

[21]

M. Schatzman, The geometry of continuous Glimm functionals,, Lectures in Applied Mathematics, 23 (1986), 417.   Google Scholar

[22]

Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chinese Ann. Math. Ser. B, 25 (2004), 37.  doi: 10.1142/S0252959904000469.  Google Scholar

[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.  doi: 10.1016/j.na.2010.04.057.  Google Scholar

show all references

References:
[1]

S. Bianchini and A. Bressan, Vanishing viscousity solutions to nonlinear hyperbolic systems,, Ann. Math., 161 (2005), 223.  doi: 10.4007/annals.2005.161.223.  Google Scholar

[2]

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

[3]

A. Bressan, "Hyerbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem,", Oxford University Press, (2000).   Google Scholar

[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.  doi: 10.1007/s11401-004-0523-4.  Google Scholar

[5]

Wen-Rong Dai, Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity,, J. Math. Anal. Appl., 327 (2007), 188.  doi: 10.1016/j.jmaa.2006.04.014.  Google Scholar

[6]

Wen-Rong Dai, Analysis of singularities for one-dimensional quasilinear hyperbolic systems,, J. Math. Anal. Appl., 362 (2010), 72.  doi: 10.1016/j.jmaa.2009.08.035.  Google Scholar

[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.  doi: 10.1016/j.jde.2006.12.020.  Google Scholar

[8]

L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations,, Lecture Notes in Mathematics, 1256 (1987), 214.  doi: 10.1007/BFb0077745.  Google Scholar

[9]

F. John, Formation of singularities in one-dimensional nonlinear wave propagation,, Commun. Pur. Appl. Math., 27 (1974), 377.  doi: 10.1002/cpa.3160270307.  Google Scholar

[10]

De-xing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data,, Chinese Ann. Math. B, 21 (2000), 413.  doi: 10.1142/S0252959900000431.  Google Scholar

[11]

De-xing Kong, "Cauchy Problem for Quasilinear Hyperbolic Systems,", MSJ Memoirs, 6 (2000).   Google Scholar

[12]

De-xing Kong, Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems,, T. Am. Math. Soc., 354 (2002), 3155.  doi: 10.1090/S0002-9947-02-02982-3.  Google Scholar

[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.  doi: 10.1016/S0362-546X(01)00121-3.  Google Scholar

[14]

De-xing Kong and Tong Yang, Asymtotic behavior of global classical solutions of quasilinear hyperbolic systems,, Commun. Part. Diff. Eq., 28 (2003), 1203.  doi: 10.1081/PDE-120021192.  Google Scholar

[15]

Ta-Tsien Li and De-xing Kong, Global classical solutions with small amplitude for general quasi-linear hyperbolic systems,, in, (1999), 203.   Google Scholar

[16]

Ta-Tsien Li, De-Xing Kong and Yi Zhou, Global classical solutions for general quasilinear nonstrictly hyperbolic systems,, Nonlinear Studies, 3 (1996), 203.   Google Scholar

[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.  doi: 10.1080/03605309408821055.  Google Scholar

[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.  doi: 10.1016/0362-546X(95)00228-N.  Google Scholar

[19]

Tai-Ping Liu, Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equations,, J. Differ. Equations, 33 (1979), 92.  doi: 10.1016/0022-0396(79)90082-2.  Google Scholar

[20]

M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem,, Indiana U. Math. J., 34 (1985), 533.  doi: 10.1512/iumj.1985.34.34030.  Google Scholar

[21]

M. Schatzman, The geometry of continuous Glimm functionals,, Lectures in Applied Mathematics, 23 (1986), 417.   Google Scholar

[22]

Yi Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chinese Ann. Math. Ser. B, 25 (2004), 37.  doi: 10.1142/S0252959904000469.  Google Scholar

[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.  doi: 10.1016/j.na.2010.04.057.  Google Scholar

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