具非光滑初始数据的拟线性双曲组的奇性形成
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摘要
一般而言,即使初始数据充分光滑并且充分小,拟线性双曲型方程组Cauchy问题的经典解只能在时间t的一个局部范围内存在并且奇性将会形成.因此,对整体解的存在性及解的生命跨度的研究具有非常重要的理论意义和实际意义.
本文考虑具有一种非光滑初始数据的一阶齐次拟线性双曲组的Cauchy问题并且ul(0)=ur(0),ul(0)≠ur(0).假设初值满足其中1/2生命跨度的估计式.
Generally speaking, the classical solutions to the first order quasilinear hy-perbolic systems only exist in a short time even for small initial data and thesingularity will occur. Therefore it is theoretical and practical interest to studythe global existence of the classical solution and the life span to the first orderquasilinear hyperbolic systems.
In this paper, we consider the Cauchy problem of the first order homogeneousstrictly hyperbolic systems with a kind of non-smooth initial data:
which isgenuinely nonlinear:
By using the characteristic method and the uniform a priori estimate method weobtain the formation of singularities of weakly discontinuous solutions and showthe estimate of the life span.
本文考虑具有一种非光滑初始数据的一阶齐次拟线性双曲组的Cauchy问题并且ul(0)=ur(0),ul(0)≠ur(0).假设初值满足其中1/2
Generally speaking, the classical solutions to the first order quasilinear hy-perbolic systems only exist in a short time even for small initial data and thesingularity will occur. Therefore it is theoretical and practical interest to studythe global existence of the classical solution and the life span to the first orderquasilinear hyperbolic systems.
In this paper, we consider the Cauchy problem of the first order homogeneousstrictly hyperbolic systems with a kind of non-smooth initial data:
which isgenuinely nonlinear:
By using the characteristic method and the uniform a priori estimate method weobtain the formation of singularities of weakly discontinuous solutions and showthe estimate of the life span.
引文
[1] John F. Formation of singularities in one-dimensional nonlinear wave propagation[J]. Com-m. Pure App.Math,1974,27:377-405.
[2] Ho¨rmander L. The lifespan of classical solutions of nonlinear hyperbolic equation[J]. Lec-ture Notes in Math,1256, Springer,1987,214-280.
[3] Li T T. Global classical solutions for quasilinear hyperbolic systems[J]. Research in AppliedMathematics32,J. Wiley/Masson,1994.
[4] Li T T, Kong D X, Zhou Y. Global classical solutions for general quasilinear non-strictlyhyperbolic systems with decay initial data[J]. Nonliear Studies,1996,3:203-229.
[5] Kong D X. Life span of classical solutions to quasilinear hyperbolic systems with slowdecay initial data[J]. Chin. Ann.of Math,2000,21B:413-440.
[6] Li T T, Kong D X. Breakdown of classical solutions to quasilinear hyperbolic systems[J].Nonlinear Analysis,TMA,2000,40:407-437.
[7] Li T T, Kong D X. A note on blow up phenomenon of classical solutions to quasilinearhyperbolic systems[J]. Nonlinear Anal.2002,49:535-539.
[8] Li T T, Kong D X, Zhou Y. Weakly linear degeneracy and global classical solutionsfor general quasilinear hyperbolic systems[J]. Cimm. Partial Diferential Equations,1994,19:1263-1317.
[9] Xu Y M. Formation of singularities for quasilinear hyperbolic systems with characteristicswith constent multiplicity[J]. Partial Diferential Equations,2005,18:355-370.
[10] Wang L B. Formation of singularities for quasilinear hyperbolic systems with characteris-tics with constant miltiplicity[J]. Partial Dif.Eqs,2003,16:240-254.
[11] Wang L B. Breakdown of C1solution to the Cauchy problem for quasilinear hyperbolicsystems with characteristics with constant miltiplicity[J]. Communication on Pur andApplied Analysis,2003,2:77-89.
[12] Xu Y M, Zhang H J. Breakdown of classical solutions to quasilinear non-strictly hyper-bolic systems with characteristics of constant multiplicity[J]. Chin.Journal of engineeringmathematics,2006,577-585.
[13] Li T T, Wang L B. Global existence of weakly discontinuous solutions to the Cauchyproblem with a kind of non-smooth initial data quasilinear hyperbolic systems[J]. ChineseAnn.Math,2004,25B:319-334.
[14]Li T T, Wang L B. Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case[J]. Chinese Ann.Math.2006,27B:53-66.
[15]Li T T, Serre D,Zhang H. The general Riemann problem for the motion of elastic strings[J]. SIAM J.Math.Anal,1992,23:1189-1203.
[16]A.Hoshiga. The lifespan of classical solutions to quasilinear hyperbolic systems in the critical case[J]. Funkcial.Ekvac,1998,41:167-188.
[17]Li S M. Cauchy problem for general first order inhomogeneous quasilinear hyperbolic system[J]. Partial Differential Equations,2002,15:46-68.
[18]Guo F. Global weakiy discontinuous solutions to the cauchy problem with a kind of non-smooth intinal data for inhomogeneous quasilinear hyperbolic systems[J]. Chin.Journal of engineering mathematics,2007,414-430.
[19]Wang L B. A remark on the cauchy problem for quasilinear hyperbolic system with char-acteristics with constent multiplicity[J]. Fudan Journal,2001,40:633-636.
[20]Li T T, Zhou Y, Kong D X. Global classical solutions for quasilinear hyperbolic systems with decay initial data[J]. Nonlinear Analysis,TMA,1997,28:1299-1332.
[21]Li T T, Kong D X. Initial value problem for general quasilinear hyperbolic systems with characteristics with constant miltiplicity[J]. Partial Diff.Eqs.,1997,10:299-322.
[22]Li T T, Wang L B. Global classical solutions to a kind of mixed initinal-boundary value problem for quasilinear hyperbolic systems [J]. Discrete and Continuous Dynamical Sys-tems,11:2(2004).
[23]Li T T, Yu, W.C. Boundary Value Problems foe Quasilinear Hyperbolic Systems[J]. Duke University Mathematics, Series v,1985.
[24]Zhou Y. Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy [J]. Chin. Ann. Math.,25B:1(2004),37-56.
[25]Yan P. Global classical solutions with small initial total variation for quasilinear hyperbolic systems[J]. Chin.Ann.Math,2002,23B:335-348.
[26]Yong Y F. Mechanism of the formation of singularities to the goursat problem for diagonal systems with linear degenerate characteristic fields [J]. Journal of Mathematical Research and Exposition,2011,31:23-33.
[27]季小东,郑琴,何春.拟线性双曲方程的两种Blowup机制[J].解放军理工大学学报,2002,99-102.
[28]刘法贵,宋庆涛.对角型拟线性双曲方程组周期解的Blowup[J].河南科学学报,2000,221-224.
[29]刘法贵,葛云飞.拟线性双曲型方程组的奇性[J].河南师范大学学报,2008,10-12.
[30]李大潜,秦铁虎.物理学与偏微分方程[M].北京：高等教育出版社,2000,6.
[2] Ho¨rmander L. The lifespan of classical solutions of nonlinear hyperbolic equation[J]. Lec-ture Notes in Math,1256, Springer,1987,214-280.
[3] Li T T. Global classical solutions for quasilinear hyperbolic systems[J]. Research in AppliedMathematics32,J. Wiley/Masson,1994.
[4] Li T T, Kong D X, Zhou Y. Global classical solutions for general quasilinear non-strictlyhyperbolic systems with decay initial data[J]. Nonliear Studies,1996,3:203-229.
[5] Kong D X. Life span of classical solutions to quasilinear hyperbolic systems with slowdecay initial data[J]. Chin. Ann.of Math,2000,21B:413-440.
[6] Li T T, Kong D X. Breakdown of classical solutions to quasilinear hyperbolic systems[J].Nonlinear Analysis,TMA,2000,40:407-437.
[7] Li T T, Kong D X. A note on blow up phenomenon of classical solutions to quasilinearhyperbolic systems[J]. Nonlinear Anal.2002,49:535-539.
[8] Li T T, Kong D X, Zhou Y. Weakly linear degeneracy and global classical solutionsfor general quasilinear hyperbolic systems[J]. Cimm. Partial Diferential Equations,1994,19:1263-1317.
[9] Xu Y M. Formation of singularities for quasilinear hyperbolic systems with characteristicswith constent multiplicity[J]. Partial Diferential Equations,2005,18:355-370.
[10] Wang L B. Formation of singularities for quasilinear hyperbolic systems with characteris-tics with constant miltiplicity[J]. Partial Dif.Eqs,2003,16:240-254.
[11] Wang L B. Breakdown of C1solution to the Cauchy problem for quasilinear hyperbolicsystems with characteristics with constant miltiplicity[J]. Communication on Pur andApplied Analysis,2003,2:77-89.
[12] Xu Y M, Zhang H J. Breakdown of classical solutions to quasilinear non-strictly hyper-bolic systems with characteristics of constant multiplicity[J]. Chin.Journal of engineeringmathematics,2006,577-585.
[13] Li T T, Wang L B. Global existence of weakly discontinuous solutions to the Cauchyproblem with a kind of non-smooth initial data quasilinear hyperbolic systems[J]. ChineseAnn.Math,2004,25B:319-334.
[14]Li T T, Wang L B. Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case[J]. Chinese Ann.Math.2006,27B:53-66.
[15]Li T T, Serre D,Zhang H. The general Riemann problem for the motion of elastic strings[J]. SIAM J.Math.Anal,1992,23:1189-1203.
[16]A.Hoshiga. The lifespan of classical solutions to quasilinear hyperbolic systems in the critical case[J]. Funkcial.Ekvac,1998,41:167-188.
[17]Li S M. Cauchy problem for general first order inhomogeneous quasilinear hyperbolic system[J]. Partial Differential Equations,2002,15:46-68.
[18]Guo F. Global weakiy discontinuous solutions to the cauchy problem with a kind of non-smooth intinal data for inhomogeneous quasilinear hyperbolic systems[J]. Chin.Journal of engineering mathematics,2007,414-430.
[19]Wang L B. A remark on the cauchy problem for quasilinear hyperbolic system with char-acteristics with constent multiplicity[J]. Fudan Journal,2001,40:633-636.
[20]Li T T, Zhou Y, Kong D X. Global classical solutions for quasilinear hyperbolic systems with decay initial data[J]. Nonlinear Analysis,TMA,1997,28:1299-1332.
[21]Li T T, Kong D X. Initial value problem for general quasilinear hyperbolic systems with characteristics with constant miltiplicity[J]. Partial Diff.Eqs.,1997,10:299-322.
[22]Li T T, Wang L B. Global classical solutions to a kind of mixed initinal-boundary value problem for quasilinear hyperbolic systems [J]. Discrete and Continuous Dynamical Sys-tems,11:2(2004).
[23]Li T T, Yu, W.C. Boundary Value Problems foe Quasilinear Hyperbolic Systems[J]. Duke University Mathematics, Series v,1985.
[24]Zhou Y. Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy [J]. Chin. Ann. Math.,25B:1(2004),37-56.
[25]Yan P. Global classical solutions with small initial total variation for quasilinear hyperbolic systems[J]. Chin.Ann.Math,2002,23B:335-348.
[26]Yong Y F. Mechanism of the formation of singularities to the goursat problem for diagonal systems with linear degenerate characteristic fields [J]. Journal of Mathematical Research and Exposition,2011,31:23-33.
[27]季小东,郑琴,何春.拟线性双曲方程的两种Blowup机制[J].解放军理工大学学报,2002,99-102.
[28]刘法贵,宋庆涛.对角型拟线性双曲方程组周期解的Blowup[J].河南科学学报,2000,221-224.
[29]刘法贵,葛云飞.拟线性双曲型方程组的奇性[J].河南师范大学学报,2008,10-12.
[30]李大潜,秦铁虎.物理学与偏微分方程[M].北京：高等教育出版社,2000,6.