一阶拟线性双曲组的整体弱间断解
|
摘要
本文系统地研究了一阶拟线性双曲型方程组定解问题的弱间断解，分别对柯西问题和混合初边值问题证明了弱间断解的整体存在性。
本文的具体安排如下：
在第一章，简单介绍了有关一阶拟线性双益组经典解的研究现状和本文的主要结果。
为方便起见，在第二章给出了一些预备知识，包括标准化坐标、广义标准化坐标、弱线性退化和弱间断解这些概念以及波的分解公式。
在第三章，考虑了具常重特征的一阶拟线性双曲组具一类非光滑初值的柯西问题，给出了此问题存在唯一的整体弱间断解的充要条件，并将所得结果应用到一般的弹性弦运动方程组和Minkowski空间R~(1+(n+1))中的时向极值曲面方程。
在第四章，研究了一阶非齐次拟线性双曲组具一类非光滑初值的柯西问题。在非齐次项满足匹配条件的假设之下，给出了此问题存在唯一的整体弱间断解的充要条件。
第五章和第六章主要研究一阶拟线性双曲组的混合初边值问题，所考虑的边界条件是一般形式的非线性边界条件。
在第五章，考虑齐次方程组的混合初边值问题。在初始和边界数据满足“小而衰减”条件下，证明了整体弱间断解的存在唯一性，并给出了主要结果在弹性弦运动方程组的混合初边值问题中的应用。
第六章是前一章的继续。本章考虑的是非齐次方程组的混合初边值问题。假设非齐次项满足匹配条件，通过一个引理简化了波的分解公式，进而证明了此混合问题存在唯一的整体弱间断解。
In this Ph.D. thesis, we study systematically the weakly discontinuous solutions for first order quasilinear hyperbolic systems. The existence and uniqueness of global weakly discontinuous solution are obtained for Cauchy problem and mixed initial-boundary value problem respectively.
The arrangement of the thesis is as follows:
First of all, a brief introduction on the study of classical solutions for first order quasilinear hyperbolic systems and the main results of this thesis are given in chapter 1．
For convenience, in chapter 2, we list some preliminaries, including some definitions, such as normalized coordinates, generalized normalized coordinates, weak linear degener-acy and weakly discontinuous solution, and the John's formula on the decomposition of waves.
In chapter 3, we study the Cauchy problem with a kind of non-smooth initial data for general quasilinear hyperbolic systems with characteristics with constant multiplicity. A necessary and sufficient condition is given to guarantee the existence and uniqueness of global weakly discontinuous solution for the Cauchy problem. The main result obtained can be used to the general system of the motion of an elastic string and the time-like extremal surface in Minkowski space R~(1+(n+1) ).
We consider the Cauchy problem with a kind of non-smooth initial data for first order inhomogeneous quasilinear hyperbolic systems in chapter 4．Under the assumption that the inhomogeneous term satisfies the matching condition, a necessary and sufficient condition for the global existence of weakly discontinuous solution for the Cauchy problem is obtained.
Chapter 5 and 6 are devoted to the study of the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems.
In chapter 5, we consider the mixed initial-boundary value problem for homogeneous quasilineai hyperbolic systems. If the initial and boundary data satisfy some "small and decaying " condition, it is proved that this problem admits a unique global weakly discon-tinuous solution. The main result is then applied to the mixed problem for system of the motion of an elastic string.
Chapter 6 is the continuation of the former chapter. If the inhomogeneous term satisfies the matching condition, by proving a lemma to simplify the formulas on the decomposition of waves, we get the existence of global weakly discontinuous solution to the mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems.
本文的具体安排如下：
在第一章，简单介绍了有关一阶拟线性双益组经典解的研究现状和本文的主要结果。
为方便起见，在第二章给出了一些预备知识，包括标准化坐标、广义标准化坐标、弱线性退化和弱间断解这些概念以及波的分解公式。
在第三章，考虑了具常重特征的一阶拟线性双曲组具一类非光滑初值的柯西问题，给出了此问题存在唯一的整体弱间断解的充要条件，并将所得结果应用到一般的弹性弦运动方程组和Minkowski空间R~(1+(n+1))中的时向极值曲面方程。
在第四章，研究了一阶非齐次拟线性双曲组具一类非光滑初值的柯西问题。在非齐次项满足匹配条件的假设之下，给出了此问题存在唯一的整体弱间断解的充要条件。
第五章和第六章主要研究一阶拟线性双曲组的混合初边值问题，所考虑的边界条件是一般形式的非线性边界条件。
在第五章，考虑齐次方程组的混合初边值问题。在初始和边界数据满足“小而衰减”条件下，证明了整体弱间断解的存在唯一性，并给出了主要结果在弹性弦运动方程组的混合初边值问题中的应用。
第六章是前一章的继续。本章考虑的是非齐次方程组的混合初边值问题。假设非齐次项满足匹配条件，通过一个引理简化了波的分解公式，进而证明了此混合问题存在唯一的整体弱间断解。
In this Ph.D. thesis, we study systematically the weakly discontinuous solutions for first order quasilinear hyperbolic systems. The existence and uniqueness of global weakly discontinuous solution are obtained for Cauchy problem and mixed initial-boundary value problem respectively.
The arrangement of the thesis is as follows:
First of all, a brief introduction on the study of classical solutions for first order quasilinear hyperbolic systems and the main results of this thesis are given in chapter 1．
For convenience, in chapter 2, we list some preliminaries, including some definitions, such as normalized coordinates, generalized normalized coordinates, weak linear degener-acy and weakly discontinuous solution, and the John's formula on the decomposition of waves.
In chapter 3, we study the Cauchy problem with a kind of non-smooth initial data for general quasilinear hyperbolic systems with characteristics with constant multiplicity. A necessary and sufficient condition is given to guarantee the existence and uniqueness of global weakly discontinuous solution for the Cauchy problem. The main result obtained can be used to the general system of the motion of an elastic string and the time-like extremal surface in Minkowski space R~(1+(n+1) ).
We consider the Cauchy problem with a kind of non-smooth initial data for first order inhomogeneous quasilinear hyperbolic systems in chapter 4．Under the assumption that the inhomogeneous term satisfies the matching condition, a necessary and sufficient condition for the global existence of weakly discontinuous solution for the Cauchy problem is obtained.
Chapter 5 and 6 are devoted to the study of the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems.
In chapter 5, we consider the mixed initial-boundary value problem for homogeneous quasilineai hyperbolic systems. If the initial and boundary data satisfy some "small and decaying " condition, it is proved that this problem admits a unique global weakly discon-tinuous solution. The main result is then applied to the mixed problem for system of the motion of an elastic string.
Chapter 6 is the continuation of the former chapter. If the inhomogeneous term satisfies the matching condition, by proving a lemma to simplify the formulas on the decomposition of waves, we get the existence of global weakly discontinuous solution to the mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems.
引文
[1] Dai Wenrong, Asymptotic behavior of global classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chinese Annals of Mathematics, 27B(2006), 263-286.
[2] Dai Wenrong & Kong Dexing, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, Journal of Differential Equations, 235(2007), 127-165.
[3] Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 14, American Mathematical Socity, Providence, Rhode Island, 1998.
[4] H. Freistuhler, Linear degeneracy and shock waves, Mathematische Zeitschrift, 207(1991), 583-596.
[5] 谷超豪，李大潜等，数学物理方程(第二版)，高等教育出版社，北京，2002．
[6] Guo Fei, Global weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for inhomogeneous quasilinear hyperbolic systems (to appear in Chinese Journal of Engineering Mathematics).
[7] Guo Fei, Global existence of weakly discontinuous solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Applied Mathematics: A Journal of Chinese Universities, 22B(2007), 181-200.
[8] Han Weiwei, Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for inhomogeneous quasilinear hyperbolic systems (to appear in Journal of Partial Differential Equations).
[9] L. Hormander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987.
[10] L. Hormander, Lectures on Nonlinear Hyperbolic Equations, Math. Appl., vol. 26, Springer, Berlin, 1997.
[11] Akira Hoshiga, The lifespan of solutions to quasilinear hyperbolic systems in the critical case, Funkcialaj Ekvacioj, 41(1998), 167-188.
[12] A. Jefferey, Quasilinear Hyperbolic Systems and Waves, Pitman Publishing, 1976.
[13] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Communications on Pure and Applied Mathematics, 27(1974), 377-405.
[14] 孔德兴，一类非严格双曲型可化约Cauchy问题的整体经典解，复旦学报(自然科学版)，33(1994)，310-317．
[15] 孔德兴，一阶拟线性双曲型方程组的Cauchy问题，复旦学报(自然科学版)，33(1994)，705-708．
[16] Kong Dexing, Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, Tokyo: The Mathematical Society of Japan, 2000.
[17] Kong Dexing, Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Annals of Mathematics, 21B(2000), 413-440.
[18] Kong Dexing & Li Tatsien, A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Analysis, 49(2002), 535-539.
[19] Kong Dexing, Sun Qingyou & Zhou Yi, The equation for time-like extremal surfaces in Minkowski space R~(2+n), Journal of Mathematical Physics, (47)2006, 013503: 16pp.
[20] Kong Dexing, Sun Qingyou & Zhou Yi, Erratum: "The equation for time-like extremal surfaces in Minkowski space R~(2+n)[J. Math. Phys. 47, 013503](2006), Journal of Mathematical Physics, (47)2006, 069901: 1pp.
[21] Kong Dexing & Yang Tong, Asymptotic behavior of Global classical solutions of quasilinear hyperbolic systems, Communications in Partial Differential Equations, 28(2003), 1203-1220.
[22] P. D. Lax, Hyperbolic systems of conservation laws Ⅱ, Communications on Pure and Applied Mathematics, 10(1957), 537-566.
[23] Li Shumin, Cauchy problem for general first order inhomogeneous quasilinear hyperbolic system, Journal of Partial Differential Equations, 15(2002), 46-68.
[24] Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics 32, J. Wiley/Masson, 1994.
[25] Li Tatsien, Une remarque sur les coordonnees normalisees et ses applications aux systemes hyperboliques quasi lineaires, Comptes Rendus de I'Academie des Sciences, Seris I Mathematic, 331(2000), 447-452.
[26] Li Tatsien, A remark on the normalized coordinates and its applications to quasilinear hyperbolic systems, in Optimal Control and Partial Differential Equations, J. L. Menaldi et al.(eds.), IOS Press, 2001, 181-187.
[27] 李大潜，拟线性双曲组选讲，2005／2-2006／5．
[28] Li Tatsien, Exact shock reconstruction, Inverse Problem, 21(2005), 673-684.
[29] Li Tatsien & Kong Dexing, Global classical discontinuous solutions to a kind of generalized Riemann porbem for general quasilinear hyperbolic systems of conservatins laws, Communications in Partial Differential Equations, 24(1999), 801-820.
[30] Li Tatsien & Kong Dexing, Breakdown of classical solutions to quasilinear hyperbolic systems, Nonlinear Analysis, 40(2000), 407-437.
[31] Li Tatsien & Kong Dexing, Initial value problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity, Journal of Partial Differential Equations, 10(1997), 299-322.
[32] Li Tatsien, Kong Dexing & Zhou Yi, Global classical solutions for general quasilinear nonstrictly hyperbolic systems with decay initial data, Nonlinear Studies, 3(1996), 203-229.
[33] Li Tatsien & Liu Fagui, Singularity caused by eigenvalues for quasilinear hyperbolic systems, Communications in Partial Differential Equations, 28(2003), 477-503.
[34] Li Tatsien & Peng Yuejun, The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Analysis, 52(2003), 573-583.
[35] Li Tatsien & Peng Yuejun, Global C~1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, Journal of Partial Differential Equations, 16(2003), 8-17.
[36] 李大潜，秦铁虎，物理学与偏微分方程(上册，第一版)，高等教育出版社，北京，1997．
[37] 李大潜，秦铁虎，物理学与偏微分方程(下册，第一版)，高等教育出版社，北京，2000．
[38] Li Tatsien & Wang Libin, Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems, Chinese Annals of Mathematics, 25B(2004), 319-334.
[39] Li Tatsien & Wang Libin, Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems, Nonlinear Analysis, 56(2004), 961-974.
[40] Li Tatsien & Wang Libin, Global existence of piecewise C~1 solutions to the generalized Riemann problem, Journal of Hyperbolic Differential Equations, 1(2004), 329-350.
[41] Li Tatsien & Wang Libin, Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Discrete & Continuous Dynamical Systems, 12(2005), 59-78.
[42] Li Tatsien & Wang Libin, The generalized nonlinear initial-boundary Riemann problem for quasilinear hyperbolic systems of conservation laws, Nonlinear Analysis, 62(2005), 1091-1107.
[43] Li Tatsien & Wang Libin, Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case, Chinese Annals of Mathematics, 27B(2006), 53-66.
[44] Li Tatsien & Yu Wenci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.
[45] 李大潜，俞文(鱼此)，一阶拟线性双曲型方程组的柯西问题，数学进展，2(1964)，152-171．
[46] Li Tatsien, Zhou Yi & Kong Dexing, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Communications in Partial Differential Equations, 19(1994), 1263-1317.
[47] Li Tatsien, Zhou Yi & Kong Dexing, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Analysis, 28(1997), 1299-1322.
[48] Liu Jianli & Zhou Yi, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Mathematical Methods in the Applied Sciences, 30(2007), 479-500.
[49] Liu Taiping, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equation, Journal of Differential Equations, 33(1979), 92-111.
[50] Liu Taiping, Quasilinear hyperbolic systems, Communications in Mathematical Physics, 68(1979), 141-172.
[51] 秦铁虎，一阶拟线性偏微分方程Cauchy问题整体光滑解的存在性，复旦学报(自然科学版)，22(1983)，41-47．
[52] Qin Tiehu, Global smooth solutions of dissipative boundary value problem for first order quasilinear hyperbolic system, Chinese Annals of Mathematics, 6B(1985), 289-298.
[53] Wang Libin, Formation of singularities for a kind of quasilinear non-strictly hyperbolic systems, Chinese Annals of Mathematics, 23B(2002), 439-454.
[54] 王利彬，关于具有常重特征的拟线性双曲组Cauchy问题的一个注记，复旦学报(自然科学版)，40(2001)，633-636．
[55] 武佩霞，一阶非齐次拟线性双曲组的柯西问题整体经典解的存在性，数学年刊，27A(2006)，93-108．
[56] 徐玉梅，拟线性双曲组经典解的奇性形成(博士论文)．
[57] Xu Yumei, Breakdown of classical solutions to quasilinear hyperbolic systems, Applied Mathematics: A Journal of Chinese Universities, 21B(2006), 437-453.
[58] 虞锦国，赵彦淳，一阶拟线性双曲型方程组的解的正规性，数学年刊，6A(1985)，595-609．
[59] Zhou Yi, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Annals of Mathematics, 25B(2004), 37-56.
[2] Dai Wenrong & Kong Dexing, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, Journal of Differential Equations, 235(2007), 127-165.
[3] Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 14, American Mathematical Socity, Providence, Rhode Island, 1998.
[4] H. Freistuhler, Linear degeneracy and shock waves, Mathematische Zeitschrift, 207(1991), 583-596.
[5] 谷超豪，李大潜等，数学物理方程(第二版)，高等教育出版社，北京，2002．
[6] Guo Fei, Global weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for inhomogeneous quasilinear hyperbolic systems (to appear in Chinese Journal of Engineering Mathematics).
[7] Guo Fei, Global existence of weakly discontinuous solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Applied Mathematics: A Journal of Chinese Universities, 22B(2007), 181-200.
[8] Han Weiwei, Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for inhomogeneous quasilinear hyperbolic systems (to appear in Journal of Partial Differential Equations).
[9] L. Hormander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987.
[10] L. Hormander, Lectures on Nonlinear Hyperbolic Equations, Math. Appl., vol. 26, Springer, Berlin, 1997.
[11] Akira Hoshiga, The lifespan of solutions to quasilinear hyperbolic systems in the critical case, Funkcialaj Ekvacioj, 41(1998), 167-188.
[12] A. Jefferey, Quasilinear Hyperbolic Systems and Waves, Pitman Publishing, 1976.
[13] F. John, Formation of singularities in one-dimensional nonlinear wave propagation, Communications on Pure and Applied Mathematics, 27(1974), 377-405.
[14] 孔德兴，一类非严格双曲型可化约Cauchy问题的整体经典解，复旦学报(自然科学版)，33(1994)，310-317．
[15] 孔德兴，一阶拟线性双曲型方程组的Cauchy问题，复旦学报(自然科学版)，33(1994)，705-708．
[16] Kong Dexing, Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, Tokyo: The Mathematical Society of Japan, 2000.
[17] Kong Dexing, Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Annals of Mathematics, 21B(2000), 413-440.
[18] Kong Dexing & Li Tatsien, A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Analysis, 49(2002), 535-539.
[19] Kong Dexing, Sun Qingyou & Zhou Yi, The equation for time-like extremal surfaces in Minkowski space R~(2+n), Journal of Mathematical Physics, (47)2006, 013503: 16pp.
[20] Kong Dexing, Sun Qingyou & Zhou Yi, Erratum: "The equation for time-like extremal surfaces in Minkowski space R~(2+n)[J. Math. Phys. 47, 013503](2006), Journal of Mathematical Physics, (47)2006, 069901: 1pp.
[21] Kong Dexing & Yang Tong, Asymptotic behavior of Global classical solutions of quasilinear hyperbolic systems, Communications in Partial Differential Equations, 28(2003), 1203-1220.
[22] P. D. Lax, Hyperbolic systems of conservation laws Ⅱ, Communications on Pure and Applied Mathematics, 10(1957), 537-566.
[23] Li Shumin, Cauchy problem for general first order inhomogeneous quasilinear hyperbolic system, Journal of Partial Differential Equations, 15(2002), 46-68.
[24] Li Tatsien, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics 32, J. Wiley/Masson, 1994.
[25] Li Tatsien, Une remarque sur les coordonnees normalisees et ses applications aux systemes hyperboliques quasi lineaires, Comptes Rendus de I'Academie des Sciences, Seris I Mathematic, 331(2000), 447-452.
[26] Li Tatsien, A remark on the normalized coordinates and its applications to quasilinear hyperbolic systems, in Optimal Control and Partial Differential Equations, J. L. Menaldi et al.(eds.), IOS Press, 2001, 181-187.
[27] 李大潜，拟线性双曲组选讲，2005／2-2006／5．
[28] Li Tatsien, Exact shock reconstruction, Inverse Problem, 21(2005), 673-684.
[29] Li Tatsien & Kong Dexing, Global classical discontinuous solutions to a kind of generalized Riemann porbem for general quasilinear hyperbolic systems of conservatins laws, Communications in Partial Differential Equations, 24(1999), 801-820.
[30] Li Tatsien & Kong Dexing, Breakdown of classical solutions to quasilinear hyperbolic systems, Nonlinear Analysis, 40(2000), 407-437.
[31] Li Tatsien & Kong Dexing, Initial value problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity, Journal of Partial Differential Equations, 10(1997), 299-322.
[32] Li Tatsien, Kong Dexing & Zhou Yi, Global classical solutions for general quasilinear nonstrictly hyperbolic systems with decay initial data, Nonlinear Studies, 3(1996), 203-229.
[33] Li Tatsien & Liu Fagui, Singularity caused by eigenvalues for quasilinear hyperbolic systems, Communications in Partial Differential Equations, 28(2003), 477-503.
[34] Li Tatsien & Peng Yuejun, The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Analysis, 52(2003), 573-583.
[35] Li Tatsien & Peng Yuejun, Global C~1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, Journal of Partial Differential Equations, 16(2003), 8-17.
[36] 李大潜，秦铁虎，物理学与偏微分方程(上册，第一版)，高等教育出版社，北京，1997．
[37] 李大潜，秦铁虎，物理学与偏微分方程(下册，第一版)，高等教育出版社，北京，2000．
[38] Li Tatsien & Wang Libin, Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems, Chinese Annals of Mathematics, 25B(2004), 319-334.
[39] Li Tatsien & Wang Libin, Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for quasilinear hyperbolic systems, Nonlinear Analysis, 56(2004), 961-974.
[40] Li Tatsien & Wang Libin, Global existence of piecewise C~1 solutions to the generalized Riemann problem, Journal of Hyperbolic Differential Equations, 1(2004), 329-350.
[41] Li Tatsien & Wang Libin, Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Discrete & Continuous Dynamical Systems, 12(2005), 59-78.
[42] Li Tatsien & Wang Libin, The generalized nonlinear initial-boundary Riemann problem for quasilinear hyperbolic systems of conservation laws, Nonlinear Analysis, 62(2005), 1091-1107.
[43] Li Tatsien & Wang Libin, Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case, Chinese Annals of Mathematics, 27B(2006), 53-66.
[44] Li Tatsien & Yu Wenci, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.
[45] 李大潜，俞文(鱼此)，一阶拟线性双曲型方程组的柯西问题，数学进展，2(1964)，152-171．
[46] Li Tatsien, Zhou Yi & Kong Dexing, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Communications in Partial Differential Equations, 19(1994), 1263-1317.
[47] Li Tatsien, Zhou Yi & Kong Dexing, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Analysis, 28(1997), 1299-1322.
[48] Liu Jianli & Zhou Yi, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Mathematical Methods in the Applied Sciences, 30(2007), 479-500.
[49] Liu Taiping, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equation, Journal of Differential Equations, 33(1979), 92-111.
[50] Liu Taiping, Quasilinear hyperbolic systems, Communications in Mathematical Physics, 68(1979), 141-172.
[51] 秦铁虎，一阶拟线性偏微分方程Cauchy问题整体光滑解的存在性，复旦学报(自然科学版)，22(1983)，41-47．
[52] Qin Tiehu, Global smooth solutions of dissipative boundary value problem for first order quasilinear hyperbolic system, Chinese Annals of Mathematics, 6B(1985), 289-298.
[53] Wang Libin, Formation of singularities for a kind of quasilinear non-strictly hyperbolic systems, Chinese Annals of Mathematics, 23B(2002), 439-454.
[54] 王利彬，关于具有常重特征的拟线性双曲组Cauchy问题的一个注记，复旦学报(自然科学版)，40(2001)，633-636．
[55] 武佩霞，一阶非齐次拟线性双曲组的柯西问题整体经典解的存在性，数学年刊，27A(2006)，93-108．
[56] 徐玉梅，拟线性双曲组经典解的奇性形成(博士论文)．
[57] Xu Yumei, Breakdown of classical solutions to quasilinear hyperbolic systems, Applied Mathematics: A Journal of Chinese Universities, 21B(2006), 437-453.
[58] 虞锦国，赵彦淳，一阶拟线性双曲型方程组的解的正规性，数学年刊，6A(1985)，595-609．
[59] Zhou Yi, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Annals of Mathematics, 25B(2004), 37-56.