拟线性双曲平衡律方程组解的整体存在性
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摘要
本文的主要目的是研究拟线性双曲方程组整体解的存在唯一性.本文的主要内容由以下几章组成:
第一章为绪言,在本章中我们对一阶拟线性双曲方程组Cauchy问题和Riemann问题的物理背景和研究现状做了一个简单介绍,并对本文要研究的几个问题加以阐述,叙述我们得到的结果.
在第二章中,我们研究了具有弱线性退化特征的双曲平衡律方程组的Cauchy问题的整体经典解的存在性.证明了:如果初值适当的小,非齐次项或者满足强耗散条件,或者满足匹配条件,那么存在唯一的整体经典解.
在第三章中,我们研究了一阶拟线性双曲守恒律方程组Riemann问题解的整体存在性.我们证明了,在第一象限内的混合初边值问题,特征要么是真正非线性的,要么是线性退化的,如果初值适当的小且满足一定的条件,则C1分片光滑解是整体唯一存在的,这个解的整体结构稳定,且相似于对应Riemann问题的解,当且仅当这个解仅包含激波和切触间断,而没有中心稀疏波或其它类型的弱间断.
This thesis concerns with the global existence of smooth solu-tions of quasilinear hyperbolic systems of balance laws. The thesis isorganized as follows.
Chapter 1 is an introduction. In this chapter, we simply recallthe physical background and present situation of the study on theCauchy problem and Riemann problem of quasilinear hyperbolic sys-tems of first order. We illustrate the problems which we shall discussand state the main results contained.
In Chapter 2, we consider a kind of quasilinear hyperbolic sys-tems with inhomogeneous terms satisfying dissipative condition ormatching condition. For the Cauchy problem of this kind of systems,we prove that, if the system is weakly linearly degenerate and theinitial data is small and satisfies some decay condition, then theCauchy problem admits a unique global classical solution on t≥0.
In Chapter 3, we investigate a class of mixed initial-boundaryvalue problems for a kind of n×n quasilinear hyperbolic systems ofconservation laws on the quarter plan. We show that the structure of the piecewise C~1 solution u = u(t,x) of the problem, which can beregarded as a perturbation of the corresponding Riemann problem, isglobally similar to that of the solution u = U(z/t) of the correspondingRiemann problem. The piecewise C~1 solution u = u(t,x) to this kindof problems is globally structure stable if and only if it contains onlynon-degenerate shocks and contact discontinuities, but no rarefactionwaves and other weak discontinuities.
第一章为绪言,在本章中我们对一阶拟线性双曲方程组Cauchy问题和Riemann问题的物理背景和研究现状做了一个简单介绍,并对本文要研究的几个问题加以阐述,叙述我们得到的结果.
在第二章中,我们研究了具有弱线性退化特征的双曲平衡律方程组的Cauchy问题的整体经典解的存在性.证明了:如果初值适当的小,非齐次项或者满足强耗散条件,或者满足匹配条件,那么存在唯一的整体经典解.
在第三章中,我们研究了一阶拟线性双曲守恒律方程组Riemann问题解的整体存在性.我们证明了,在第一象限内的混合初边值问题,特征要么是真正非线性的,要么是线性退化的,如果初值适当的小且满足一定的条件,则C1分片光滑解是整体唯一存在的,这个解的整体结构稳定,且相似于对应Riemann问题的解,当且仅当这个解仅包含激波和切触间断,而没有中心稀疏波或其它类型的弱间断.
This thesis concerns with the global existence of smooth solu-tions of quasilinear hyperbolic systems of balance laws. The thesis isorganized as follows.
Chapter 1 is an introduction. In this chapter, we simply recallthe physical background and present situation of the study on theCauchy problem and Riemann problem of quasilinear hyperbolic sys-tems of first order. We illustrate the problems which we shall discussand state the main results contained.
In Chapter 2, we consider a kind of quasilinear hyperbolic sys-tems with inhomogeneous terms satisfying dissipative condition ormatching condition. For the Cauchy problem of this kind of systems,we prove that, if the system is weakly linearly degenerate and theinitial data is small and satisfies some decay condition, then theCauchy problem admits a unique global classical solution on t≥0.
In Chapter 3, we investigate a class of mixed initial-boundaryvalue problems for a kind of n×n quasilinear hyperbolic systems ofconservation laws on the quarter plan. We show that the structure of the piecewise C~1 solution u = u(t,x) of the problem, which can beregarded as a perturbation of the corresponding Riemann problem, isglobally similar to that of the solution u = U(z/t) of the correspondingRiemann problem. The piecewise C~1 solution u = u(t,x) to this kindof problems is globally structure stable if and only if it contains onlynon-degenerate shocks and contact discontinuities, but no rarefactionwaves and other weak discontinuities.
引文
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[2] Courant R. and Friedrichs K. O., Supersonic ?ow and shock waves, Berlin,Springer, (1976).
[3] Chorin A. J. and Marsden J., A Mathematical Introduction to FluidMechanics, New York, Springer-Verlag, (1979).
[4] Landau L. D. and Lifschitz E. M., Fluid Mechanics, Addison Wesley,Reading MA, (1971).
[5] Serrin J. , Mathematical Principles of Classical Fluid Mechanics, inHandbuch der Physik 8, Springer-Verlag, (1959).
[6] Taub A. H., Relativistic ?uid mechanics, Annu. Rev. Fluid Mech. (1978),10, 301-332.
[7] 姜礼尚、孔德兴、陈志浩,应用偏微分方程讲义,北京, 高等教育出版社出版,(2008).
[8] 李大潜、陈韵梅, 非线性发展方程, 2, 北京, 科学出版社, (1997).
[9] 戴文荣, 一阶拟线性双曲型方程组整体经典解的渐进性态与奇性分析, [学位论文], 上海交通大学图书馆, 上海交通大学, (2005).
[10] John F., Formation of singularities in one-dimensional nonlinear wavepropagation, Comm. Pure Appl. Math., (1974), 27, 377-405 .
[11] Liu T. P., Development of singularities in the nonlinear waves for quasilinearhyperbolic partial di?erential equations, J. Di?. Eqs. (1979), 33, 92-111.
[12] L. H¨ormander, The lifespan of classical solutions of nonlinear hyperbolicequations, Lecture Notes in Math. 1256, Springer, (1987), 214-280.
[13] Li T. T., Zhou Y. and Kong D. X., Weak linear degeneracy and the globalclassical solutions for general quasilinear hyperbolic systems, Comm. inPartial Di?erential Equations, (1994), 19, 1263-1317 .
[14] Li T. T., Zhou Y. and Kong D. X., Global classical solutions for generalquasilinear hyperbolic systems with decay initial data, Nonlinear AnalysisTMA, (1997), 28, 1299-1332.
[15] Li T. T. and Kong D. X., Breakdown of classical solutions to quasilinearhyperbolic systems, Nonlinear Analysis, Theory, Method and Applications,(2000), 40, 407-437.
[16] Kong D. X., Cauchy Problem for Quasiliear Hyperbolic Systems, Vol. 6Tokyo, MSJ Memoirs, (2000).
[17] Kong D. X. and Yang T., Asymtotic behavior of global classical solutionsof quasilinear hyperbolic systems, Comm. in Partial Di?erential Equations,(2003), 28, 1203-1220.
[18] Kong D. X., Life-span of classical solutions to quasilinear hyperbolic systemswith slow decay initial data, Chinese Ann. of Math., (2000), 21B, 413-440 .
[19] Weiland J. C. et Wilhelmsson H., Coherent Non Linear Interaction of Wavesin Plasmas, Pergamon Press, (1977).
[20] Kaup D. J., Reiman A. et Bers A., Space-time evolution of non-linearthree-wave interaction I: Interaction in a homogeneous medium, Reviewsof Modern Physics, (1979), 51, 275-309.
[21] Bamberger A., Flytzanis C. et Jennev′e D., E′tude math′ematique etnum′erique d’un mod`ele de propagation d’ondes lumineuses dans unefibre optique non centrosym′etrique, E′cole Polytechnique, Math′ematiquesappliqu′ees, (1986), 149.
[22] Kong D. X., Maximum principle in nonlinear hyperbolic systems and itsapplications, Nonlinear Analysis, Theory, Method & Applications, (1998),32, 871-880 .
[23] Hsiao L. and Li T. T., Global smooth solution of Cauchy problems for a classof quasilinear hyperbolic systems, Chin. Ann. of Math., (1983), 4B, 109-115.
[24] Li T. T. and Qin T. H., Global smooth solutions for a class of quasilinearhyperbolic systems with dissipative terms, Chin. Ann. of Math., (1985), 6B,199-210.
[25] Kosin′ski W., Gradient catastrophe in the solution of nonconservativehyperbolic systems, J. Math. Anal. Appl., (1977) 61, 672-688.
[26] Kong D. X., Cauchy problem for quasilinear hyperbolic systems with higherorder dissipative terms, Nonlinear Di?erential Equations and Applications,(1997), 4, 477-489.
[27] Smoller J., Shock waves and reaction-di?usion equations, Second Edition, ASeries of Comprehensive Studies in Mathematics, Springer-Verlag, (1999).
[28] Lax P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl.Math., (1957), 10, 537-556.
[29] Kong D. X., Global structure stability of Riemann solutions of quasilinearhyperbolic systems of conservation laws: shocks and contact discontinuities,J. Di?erential Equations, (2003), 188, 242-271.
[30] Kong D. X., Global structure instability of Riemann solutions of quasilinearhyperbolic systems of conservation laws: rarefraction waves, Journal of Di?.Equa.,(2005), 219, 421-450.
[31] Shao Z. Q., Global structure instability of Riemann solutions for generalquasilinear hyperbolic systems of conservation laws in the presence of aboundary, Journal of Mathematical Analysis and Applications, (2007), 330,511-540 .
[32] Shao Z. Q., Global structure stability of Riemann solutions for hyperbolicsystems of conservation laws in the presence of a boundary corresponding tocontact discontinuities, to appear.
[33] Kruzkov S.N., Generalized solutions of the Cauchy problem in the largefor first order nonlinear equations (Russian), Dokl. Akad. Nauk. SSSR 187(1969), 29-32.
[34] Glimm J., Solutions in the large for nonlinear hyperbolic systems ofequations, Comm. Pure Appl. Math., (1965), 18, 697-715.
[35] DiPerna R., Decay and asymptotic behavior of solutions to nonlinearhyperbolic systems of conservation laws, Ind. U. Math. J., (1975), 24, 1047-1071.
[36] DiPerna R., Decay of solutions to nonlinear hyperbolic systems of conser-vation laws with a convex extension, Arch. Rat. Mech. Anal., (1977), 60,1-46.
[37] Liu T. P., Decay to N-waves of solutions of general systems of nonlinearhyperbolic conservation laws, Comm. Pure and Appl. Math., (1977), XXX,585-610.
[38] Liu T. P., Linear and nonlinear large-time behavior of solutions of generalsystems of hyperbolic conservation laws, Comm. Pure Appl. Math., (1977),30, 767-796.
[39] Greenberg J., Asymptotic behavior of solutions to the quasilinear waveequations, Partial di?erential equations and related topics, Springer LectureNotes in Math., (1975), 446 198-246.
[40] Bressan A., Contractive metrices for nonlinear hyperbolic systems, IndianaUniv. Math. J., (1988), 37, 409-421.
[41] Bressan A., The unique limit of the Glimm scheme, Arch. Rational Mech.Anal., (1995), 130, 1-75.
[42] Bressan A., Global solutions of systems of conservation laws by wave fronttracking, J. Math. Anal. Appl., (1992), 170, 414-432.
[43] Bressan A., Analysis of solutions to hyperbolic systems by the front trackingmethod, Monographs and surveys in Pure and Appl. Math. 99, Editor H.Freishu¨ler, Chapman and Hall, Analysis of systems of conservation laws,(1997).
[44] Liu T. P. and Yang T., Uniform L1 boundness of solutions of hyperbolicconservation laws, Methods and Applications of Analysis, (1997), 4, 339-355.
[45] Liu T. P. and Yang T., Well-posedness theory for hyperbolic conservationlaws, Comm. Pure Appl. Math., (1999), 52, No.12, 1553-1586.
[46] Liu T. P. and Yang T., L1 stablity of conservation laws with coincidingHugoniot and characteristic curves, Indiana Univ. Math. J., (1999), 48,No.1, 237-248.
[47] Liu T. P. and Yang T., Weak solutions of general system of hyperbolicconservation laws, Comm. Math. Phys., (2002), 2, 289-327.
[48] Kato T., The Cauchy problem for quasilinear symmetric hyperbolic systems,Arch. Rat. Mech. Anal., (1975), 58, 181-205.
[49] Klainerman S. and Majda A., Formation of singularities for wave equationsincluding the nonlinear vibrating string, Communications on Pure andApplied Mathematics, (1980), 33, 241-263.
[50] Alinhac S., Blowup for nonlinear hyperbolic equations, Boston, Birkhauser,(1995).
[51] Sideris T. C., Formation of singularities in three-dimensional compressible?uids, Comm. Math. Phys., (1985), 101, 475-485.
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