双层浅水波模型柯西问题的经典解

刘见礼, 张小丹

数学学报 ›› 2015, Vol. 58 ›› Issue (6) : 985-992.

PDF(485 KB)
中国科学院数学与系统科学研究院期刊网
PDF(485 KB)
数学学报 ›› 2015, Vol. 58 ›› Issue (6) : 985-992. DOI: 10.12386/A2015sxxb0097
论文

双层浅水波模型柯西问题的经典解

    刘见礼, 张小丹
作者信息 +

Classical Solutions to Cauchy Problem for the Model of Two-Layer Flows

    Jian Li LIU, Xiao Dan ZHANG
Author information +
文章历史 +

摘要

主要研究浅水理论中双层浅水波模型柯西问题的经典解.在适当的初值条件下, 得到整体经典解存在唯一性的充要条件.同时对单层浅水波模型也得到相应结果.

Abstract

We consider classical solutions to Cauchy problem for the model of twolayer flows, which appears in shallow water theory. Under the appropriate assumptions on initial data, we can obtain the necessary and sufficient conditions of the global existence of C1 solutions. Furthermore, similar conclusion can be obtained for the system of one-layer flow.

关键词

柯西问题 / 经典解 / 浅水波方程

Key words

Cauchy problem / classical solutions / shallow water equations

引用本文

EndNote

Ris (Procite)

Bibtex

导出引用
刘见礼, 张小丹. 双层浅水波模型柯西问题的经典解. 数学学报, 2015, 58(6): 985-992 https://doi.org/10.12386/A2015sxxb0097
Jian Li LIU, Xiao Dan ZHANG. Classical Solutions to Cauchy Problem for the Model of Two-Layer Flows. Acta Mathematica Sinica, Chinese Series, 2015, 58(6): 985-992 https://doi.org/10.12386/A2015sxxb0097

参考文献

[1] Bleck R., Ocean modeling in isopycnic coordinates, Ocean Modeling and Parameterization, Kluwer Academic, 1998, 516: 423–448.

[2] Chen L. L., Tian L. X., On the weak solution to the general shallow water wave equation, Int. J. Nonlinear Sci., 2009, 8(2): 194–200.

[3] Camassa R., Holm D. D., Hyman J. M., A new integrable shallow water equation, Adv. Appl. Mech., 1994, 31: 1–33.

[4] Constantin A., Escher J., Well-posedness, global existence and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 1998, 51(5): 475–504.

[5] Constantin A., Escher J., On the Cauchy problem for a family of quasilinear hyperbolic equations, Comm. Partial Diff. Equ., 1998, 23(7-8): 1449–1458.

[6] Constantin A., Escher J., Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1998, 26(2): 303–328.

[7] Constantin A., Escher J., Global weak solutions for a shallow water equation, Indiana Univ. Math. J., 1998, 47(4): 1527–1545.

[8] Constantin A., Escher J., On the structure of a family of quasilinear equations arising in shallow water theory, Math. Ann., 1998, 312(3): 403–416.

[9] Chumakova L., Menzaque F. E., Milewski P. A., et al., Stability properties and nonlinear mappings of two and three-layer stratified flows, Stud. Appl. Math., 2009, 122(2): 123–137.

[10] Fu X. Y., Sharma V. D., Cauchy problem for quasilinear hyperbolic systems of shallow water equations, Appl. Anal., 2013, 92(11): 2309–2319.

[11] Guo B. L., Tian L. X., Yang L. E, et al., Camassa-Holm Equation, Science Press, Beijing, 2008 (in Chinese).

[12] Jeffrey A., Quasilinear Hyperbolic Systems and Waves, Research Notes in Mathematics, London, 1976.

[13] Karelsky K. V., Petrosyan A. S., Particular solutions and Riemann problem for modified shallow water equations, Fluid Dynam. Res., 2006, 38(5): 339–358.

[14] Kong, D. X., Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, Mathematical Society of Japan, Tokyo, 2000.

[15] Li T. T., Global Classical Solutions of Quasilinear Hyperbolic Systems, Research in Applied Mathematics, 32, Wiley/masson, New York, 1994.

[16] Lax P. D., Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 1964, 5: 611–613.

[17] Li T. T., Zhou Y., Kong D. X., Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Diff. Equ., 1994, 19(7–8): 1263–1317.

[18] Li T. T., Zhou Y., Kong D. X., Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal., 1997, 28(8): 1299–1332.

[19] Li T. T., Wang L. B., Global propagation of regular nonlinear hyperbolic waves, Progress in Nonlinear Differential Equations and their Applications, 76. Birkhäuser Boston, Inc., Boston, MA, 2009.

[20] Milewski P. A., Tabak E. G., Turner C. V., Nonlinear stability of two-layer flows, Commun. Math. Sci., 2004, 2(3): 427–442.

[21] Marcati P., Rubino B., Hyperbolic to parabolic relaxation theory for quasilinear first order systems, J. Diff. Equ., 2000, 162(2): 359–399.

[22] Sekhar T. R., Sharma V. D., Interaction of shallow water waves, Stud. Appl. Math., 2008, 121(1): 1–25.

[23] Tian L. X., Zhang P., Xia L. M., Global existence for the higher-order Camassa–Holm shallow water equation, Nonlinear Anal., 2011, 74(7): 2468–2474.

[24] Tian L. X., Gui G. L., Liu Y., On the Cauchy problem for the generalized shallow water wave equation, J. Differential Equations, 2008, 245(7): 1838–1852.

[25] Tian L. X., Li X. M., Well-posedness for a new completely integrable shallow water wave equation, Int. J. Nonlinear Sci., 2007, 4(2): 83–91.

[26] Yang T., Zhu C. J., Existence and non-existence of global smooth solutions for p-system with relaxation, J. Diff. Equ., 2000, 161(2): 321–336.

[27] Zilitinkevich S. S., Chalikov D. V., Resnyansky Y. D., Modeling the oceanic upper layer, Oceanol. Acta, 1979, 2(2): 219–240.

[28] Zhou Y., Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B, 2004, 25(1): 37–56.

基金

国家自然科学基金资助项目(11401367);教育部博士点基金资助项目(20133108120002)

PDF(485 KB)

Accesses

Citation

Detail
段落导航
相关文章

/

〈 〉