一阶拟线性双曲组的奇性形成机制
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摘要
在此博士论文中，我们深入研究了一阶拟线性双曲型方程组的奇性形成机制．
首先，在第一章简要介绍一阶拟线性双曲组奇性形成机制的研究背景及现状．
为了方便，我们在第二章罗列了一些预备知识，包括一些定义，例如标准化坐标，弱线性退化和弱间断解，以及F.John的波分解公式．
人们通常认为对具线性退化特征场的(守恒律)拟线性双曲组不会导致激波形成．这在很长一段时间内都是一个猜想(见[4]，[42])，而且至今仍然悬而未决．在第三章，提出一个证明此猜想的一般框架，并借助于块严格双曲、部分rich性以及逐块封闭系统的概念给出了满足此猜想的一些一般的拟线性双曲组．
在第四章，我们在第一象限{(t,x)|t≥0，x≥0}上对具非线性边界条件的拟线性双曲组考虑其混合初边值问题．在系统为严格双曲以及初始数据和边界条件具有某种“小”性的假设下，分别对线性退化和弱线性退化的系统得到了唯一整体C~1解及相应的L~1稳定性，并应用于两个对应的物理模型．
在最后一章，对具线性退化特征场的非齐次对角型拟线性严格双曲组及具常重特征的双曲组，在初始数据C~1模有界的假设下，证明了其Cauchy问题的经典解的奇性必为ODE奇性．对半无界区域上相应的初边值问题、Goursat问题以及具弱间断边界条件的Goursat问题也得到了相同的结论．
In this Ph.D. thesis, we consider the mechanism of the formation of singularities for first order quasilinear hyperbolic systems.
First of all, a brief introduction of the background and the present situation on the study of mechanism of the formation of singularities for first order quasilinear hyperbolic systems is given in Chapter 1.
For convenience, in Chapter 2, we list some preliminaries, including several definitions, such as normalized coordinates, weak linear degeneracy, weakly discontinuous solution, and the John's formula on the decomposition of waves.
One often believes that there is no shock formation for the Cauchy problem of quasilinear hyperbolic systems (of conservation laws) with linearly degenerate characteristic fields. It has been a conjecture for a long time (see [4], [42]) and it is still an open problem in the general situation up to now. In Chapter 3, a framework to justify this conjecture is proposed, and, by means of the concept such as the strict block-hyperbolicity, the part richness and the successively block-closed system, some general kinds of quasilinear hyperbolic systems, which verify the conjecture, are presented.
In Chapter 4, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant {(t, x)| t≥0, x≥0}. Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence, uniqueness and L~1 stability of C~1 solutions are obtained for small initial and boundary data. We also give two applications for corresponding physical models.
In the last chapter, for inhomogeneous diagonal system with distinct characteristics or with characteristics with constant multiplicity, under the assumption that the system is linearly degenerate and the C~1 norm of the initial data is bounded, we show that the mechanism of the formation of singularities of classical solution to its Cauchy problem must be of ODE type. Similar results are also obtained for corresponding mixed initial-boundary value problems on a semi-unbounded domain, Goursat problem with C~1 boundary conditions or with weakly discontinuous boundary conditions.
首先，在第一章简要介绍一阶拟线性双曲组奇性形成机制的研究背景及现状．
为了方便，我们在第二章罗列了一些预备知识，包括一些定义，例如标准化坐标，弱线性退化和弱间断解，以及F.John的波分解公式．
人们通常认为对具线性退化特征场的(守恒律)拟线性双曲组不会导致激波形成．这在很长一段时间内都是一个猜想(见[4]，[42])，而且至今仍然悬而未决．在第三章，提出一个证明此猜想的一般框架，并借助于块严格双曲、部分rich性以及逐块封闭系统的概念给出了满足此猜想的一些一般的拟线性双曲组．
在第四章，我们在第一象限{(t,x)|t≥0，x≥0}上对具非线性边界条件的拟线性双曲组考虑其混合初边值问题．在系统为严格双曲以及初始数据和边界条件具有某种“小”性的假设下，分别对线性退化和弱线性退化的系统得到了唯一整体C~1解及相应的L~1稳定性，并应用于两个对应的物理模型．
在最后一章，对具线性退化特征场的非齐次对角型拟线性严格双曲组及具常重特征的双曲组，在初始数据C~1模有界的假设下，证明了其Cauchy问题的经典解的奇性必为ODE奇性．对半无界区域上相应的初边值问题、Goursat问题以及具弱间断边界条件的Goursat问题也得到了相同的结论．
In this Ph.D. thesis, we consider the mechanism of the formation of singularities for first order quasilinear hyperbolic systems.
First of all, a brief introduction of the background and the present situation on the study of mechanism of the formation of singularities for first order quasilinear hyperbolic systems is given in Chapter 1.
For convenience, in Chapter 2, we list some preliminaries, including several definitions, such as normalized coordinates, weak linear degeneracy, weakly discontinuous solution, and the John's formula on the decomposition of waves.
One often believes that there is no shock formation for the Cauchy problem of quasilinear hyperbolic systems (of conservation laws) with linearly degenerate characteristic fields. It has been a conjecture for a long time (see [4], [42]) and it is still an open problem in the general situation up to now. In Chapter 3, a framework to justify this conjecture is proposed, and, by means of the concept such as the strict block-hyperbolicity, the part richness and the successively block-closed system, some general kinds of quasilinear hyperbolic systems, which verify the conjecture, are presented.
In Chapter 4, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant {(t, x)| t≥0, x≥0}. Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence, uniqueness and L~1 stability of C~1 solutions are obtained for small initial and boundary data. We also give two applications for corresponding physical models.
In the last chapter, for inhomogeneous diagonal system with distinct characteristics or with characteristics with constant multiplicity, under the assumption that the system is linearly degenerate and the C~1 norm of the initial data is bounded, we show that the mechanism of the formation of singularities of classical solution to its Cauchy problem must be of ODE type. Similar results are also obtained for corresponding mixed initial-boundary value problems on a semi-unbounded domain, Goursat problem with C~1 boundary conditions or with weakly discontinuous boundary conditions.
引文
[1] Alinhac S., Blowup for nonlinear hyperbolic equations, Progress in Nonlinear D.E. and their applications , 17 (1995), Birkhauser.
[2] Aregba-Driollet Denise & Hanouzet Bernard, Cauchy problem for one-dimensional semilinear hyperbolic systems:global existence, blow up, J. Diff. Eqs 125(1996), 1-26.
[3] Boillat G., Chocs caracteristiques, C.R. Acad. Sci. Paris, Serie I, 274(1972), 1018-1021.
[4] Brenier Y., Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172(2004), 65-91.
[5] Bressan A., Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J., 37(1988), 409-421.
[6] Bressan A., Liu T.P. & Yang T., L~1 stability estimates for n × n conservation laws, Arch. Rat. Mech. Anal., 149(1999), 1-22.
[7] Carbou G. & Hanouzet B., Semilinear behaviour for a quasilinear hyperbolic system: the Kerr Debye model, C.R. Acad. Sci. Paris. Ser. I, 343(2006), 243-247.
[8] Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 14, American Mathematical Socity, Providence, Rhode Island, 1998.
[9] Greenberg J.M. & Li T.T., The effect of boundary damping for the quasilinear wave equation, J. Diff. Eqs., 52(1984), 66-75.
[10]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．
[11] Hormander L., The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture notes in Math., Springer, 1256(1987), 214-280.
[12] Jeffrey A., Breakdown of the solution to a completely exceptional systems of hyperbolic equations, J. Math. Anal. Appl., 45(1974), 375-381.
[13] A. Jefferey, Quasilinear Hyperbolic Systems and Waves, Pitman Publishing, 1976.
[14] John F., Nonlinear waves equations, formation of singularities, Pitcher Lectures in Math. Sciences, Lehigh University, American Math. Society, 1990.
[15] John F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27(1974), 377-405.
[16] Freistühler H., Rotational degeneracy of hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 113(1991), 39-64.
[17] Kong D.X., Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chin. Ann. Math., 21B (2000), 413-440.
[18] Kong D.X. & Li T.T., A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal. TMA, 49(2002), 535-539.
[19] Lax P.D., Hyperbolic systems of conservation laws II, Communications on Pure and Applied Mathematics , 10(1957), 537-566.
[20] Lax P.D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations , J. Math. Phys., 5(1964), 611-613.
[21] Li T.T., Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson/John Wiley, 1994.
[22] Li T.T., Complete reduciblility for quasilinear hyperbolic systems, J. Partial Diff. Eqs., 19(2006), 10-15.
[23]李大潜，拟线性双曲组选讲，2005／2-2006／5．
[24] Li T.T. & Kong D.X., Breakdown of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal. TMA, 40(2000), 407-437.
[25] Li T.T. & Liu F.G., Singularity caused by eigenvectors for quasilinear hyperbolic systems, Comm. Partial Diff. Eqs., 28(2003), 477-503.
[26] Li T.T. & Peng Y.J., The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal., TMA, 52(2003), 573-583.
[27] Li T.T. & Peng Y.J., Gloabal C~1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, J. Partial Diff. Eqs. 16(2003), 8-17.
[28] Li T.T. & Peng Y.J., Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form, Nonlinear Anal., 55 2003, 937-949.
[29]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．
[30]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．
[31] Li T.T. & Wang L.B., Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems, Chin. Ann. Math., 25B(2004), 319-334.
[32] Li T.T. & Wang L.B., Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Discrete & Continuous Dynamical Systems, 12(2005), 59-78.
[33]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．
[34] Li T.T., Yu W.C., Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.
[35] Li T.T., Zhou Y. & Kong D.X., Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Eqs., 19(1994), 1263-1317.
[36] Li T.T., Zhou Y. & Kong D.X., Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal., TMA, 28(1997), 1299-1232.
[37] Liu F.G. & Ye T, Global classical solutions for quasilinear hyperbolic systems in 3-step completely reducible type, Mathematica Applicata, 20(2007), 581-586.
[38]刘法贵，叶晓枫，具常数特征拟线性双曲型方程组Cauchy问题，数学研究，39(2006)，164-174．
[39] Liu T.P., Development of singularities in the nonlinear waves for quasilinear hyperbolic patial differential equations, J. Diff. Eqs., 33(1979), 92-111.
[40] Liu T.P., Quasilinear hyperbolic systems, Communications in Mathematical Physics, 68(1979), 141-172.
[41] Liu T.P. & Yang T., Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52(1999), 1553-1586.
[42] Majda A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, New York, Springer-Verlag, 1984.
[43] Qin T.H., Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems, Chin. Ann. Math., 6B(1985), 289-298.
[44] Rozdestvenskii B.L. & Yanenko N.N., Systems of Quasilinear Hyperbolic Equations, Izd. Nauka, Moskva, 1968.
[45] Schartzman M., Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana Univ. Math. J., 34(1985), 533-589.
[46] Schartzman M., Geometry of continuous Glimm functional, Lectures in Applied Mathematics, Vol. 23, 1986,417-439.
[47] Serre D., Systemes de Lois de Conservation II, Diderot: Paris, 1996.
[48] Sevennec B., Geometie des systemes de lois de conservation, Memoire Soc. Math, de France, Vol. 56, 1994.
[49] Wang L.B., Formation of singularities for a kind of quasilinear non-strictly hyperbolic systems, Chin. Ann. Math., 23B(2002), 439-454.
[50] Wang W.K. & Yang X.F., Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems, Chin. Ann. Math., 24B(2003), 457-468.
[51] Yan P., Global classical solutions with small initial total variation for quasilinear hyperbolic systems, Chin. Ann. Math., 23B(2002), 335-348.
[52] Zhou Y., Low regularity solutions for linearly degenerate hyperbolic systems, Nonlinear Anal. TMA, 26(1996), 1843-1857.
[53] Zhou Y, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chin. Ann. Math., 25B(2004), 37-56.
[2] Aregba-Driollet Denise & Hanouzet Bernard, Cauchy problem for one-dimensional semilinear hyperbolic systems:global existence, blow up, J. Diff. Eqs 125(1996), 1-26.
[3] Boillat G., Chocs caracteristiques, C.R. Acad. Sci. Paris, Serie I, 274(1972), 1018-1021.
[4] Brenier Y., Hydrodynamic structure of the augmented Born-Infeld equations, Arch. Rat. Mech. Anal., 172(2004), 65-91.
[5] Bressan A., Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J., 37(1988), 409-421.
[6] Bressan A., Liu T.P. & Yang T., L~1 stability estimates for n × n conservation laws, Arch. Rat. Mech. Anal., 149(1999), 1-22.
[7] Carbou G. & Hanouzet B., Semilinear behaviour for a quasilinear hyperbolic system: the Kerr Debye model, C.R. Acad. Sci. Paris. Ser. I, 343(2006), 243-247.
[8] Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 14, American Mathematical Socity, Providence, Rhode Island, 1998.
[9] Greenberg J.M. & Li T.T., The effect of boundary damping for the quasilinear wave equation, J. Diff. Eqs., 52(1984), 66-75.
[10]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．
[11] Hormander L., The lifespan of classical solutions of nonlinear hyperbolic equations, Lecture notes in Math., Springer, 1256(1987), 214-280.
[12] Jeffrey A., Breakdown of the solution to a completely exceptional systems of hyperbolic equations, J. Math. Anal. Appl., 45(1974), 375-381.
[13] A. Jefferey, Quasilinear Hyperbolic Systems and Waves, Pitman Publishing, 1976.
[14] John F., Nonlinear waves equations, formation of singularities, Pitcher Lectures in Math. Sciences, Lehigh University, American Math. Society, 1990.
[15] John F., Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math., 27(1974), 377-405.
[16] Freistühler H., Rotational degeneracy of hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal., 113(1991), 39-64.
[17] Kong D.X., Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chin. Ann. Math., 21B (2000), 413-440.
[18] Kong D.X. & Li T.T., A note on blow-up phenomenon of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal. TMA, 49(2002), 535-539.
[19] Lax P.D., Hyperbolic systems of conservation laws II, Communications on Pure and Applied Mathematics , 10(1957), 537-566.
[20] Lax P.D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations , J. Math. Phys., 5(1964), 611-613.
[21] Li T.T., Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson/John Wiley, 1994.
[22] Li T.T., Complete reduciblility for quasilinear hyperbolic systems, J. Partial Diff. Eqs., 19(2006), 10-15.
[23]李大潜，拟线性双曲组选讲，2005／2-2006／5．
[24] Li T.T. & Kong D.X., Breakdown of classical solutions to quasilinear hyperbolic systems, Nonlinear Anal. TMA, 40(2000), 407-437.
[25] Li T.T. & Liu F.G., Singularity caused by eigenvectors for quasilinear hyperbolic systems, Comm. Partial Diff. Eqs., 28(2003), 477-503.
[26] Li T.T. & Peng Y.J., The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal., TMA, 52(2003), 573-583.
[27] Li T.T. & Peng Y.J., Gloabal C~1 solution to the initial-boundary value problem for diagonal hyperbolic systems with linearly degenerate characteristics, J. Partial Diff. Eqs. 16(2003), 8-17.
[28] Li T.T. & Peng Y.J., Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form, Nonlinear Anal., 55 2003, 937-949.
[29]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．
[30]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．
[31] Li T.T. & Wang L.B., Global existence of weakly discontinuous solutions to the Cauchy problem with a kind of non-smooth initial data for quasilinear hyperbolic systems, Chin. Ann. Math., 25B(2004), 319-334.
[32] Li T.T. & Wang L.B., Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Discrete & Continuous Dynamical Systems, 12(2005), 59-78.
[33]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．
[34] Li T.T., Yu W.C., Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.
[35] Li T.T., Zhou Y. & Kong D.X., Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Eqs., 19(1994), 1263-1317.
[36] Li T.T., Zhou Y. & Kong D.X., Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal., TMA, 28(1997), 1299-1232.
[37] Liu F.G. & Ye T, Global classical solutions for quasilinear hyperbolic systems in 3-step completely reducible type, Mathematica Applicata, 20(2007), 581-586.
[38]刘法贵，叶晓枫，具常数特征拟线性双曲型方程组Cauchy问题，数学研究，39(2006)，164-174．
[39] Liu T.P., Development of singularities in the nonlinear waves for quasilinear hyperbolic patial differential equations, J. Diff. Eqs., 33(1979), 92-111.
[40] Liu T.P., Quasilinear hyperbolic systems, Communications in Mathematical Physics, 68(1979), 141-172.
[41] Liu T.P. & Yang T., Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math., 52(1999), 1553-1586.
[42] Majda A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, New York, Springer-Verlag, 1984.
[43] Qin T.H., Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems, Chin. Ann. Math., 6B(1985), 289-298.
[44] Rozdestvenskii B.L. & Yanenko N.N., Systems of Quasilinear Hyperbolic Equations, Izd. Nauka, Moskva, 1968.
[45] Schartzman M., Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana Univ. Math. J., 34(1985), 533-589.
[46] Schartzman M., Geometry of continuous Glimm functional, Lectures in Applied Mathematics, Vol. 23, 1986,417-439.
[47] Serre D., Systemes de Lois de Conservation II, Diderot: Paris, 1996.
[48] Sevennec B., Geometie des systemes de lois de conservation, Memoire Soc. Math, de France, Vol. 56, 1994.
[49] Wang L.B., Formation of singularities for a kind of quasilinear non-strictly hyperbolic systems, Chin. Ann. Math., 23B(2002), 439-454.
[50] Wang W.K. & Yang X.F., Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems, Chin. Ann. Math., 24B(2003), 457-468.
[51] Yan P., Global classical solutions with small initial total variation for quasilinear hyperbolic systems, Chin. Ann. Math., 23B(2002), 335-348.
[52] Zhou Y., Low regularity solutions for linearly degenerate hyperbolic systems, Nonlinear Anal. TMA, 26(1996), 1843-1857.
[53] Zhou Y, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chin. Ann. Math., 25B(2004), 37-56.