SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM

doi:10.1016/S0252-9602(13)60041-7

数学物理学报(英文版) ›› 2013, Vol. 33 ›› Issue (3): 821-829.doi: 10.1016/S0252-9602(13)60041-7

SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM

周寿明^1,2|穆春来²|王良展²

1. College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China；
2. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

收稿日期:2011-11-16 出版日期:2013-05-20 发布日期:2013-05-20
基金资助:
This work is supported by NSF of China (11071266), partially supported by Scholarship Award for Excellent Doctoral Student granted by Ministry of Education and partially supported by the found of Chongqing Normal University (13XLB006).

SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM

ZHOU Shou-Ming^1,2, MU Chun-Lai², WANG Liang-Zhan²

1. College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China；
2. College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received:2011-11-16 Online:2013-05-20 Published:2013-05-20
Supported by:
This work is supported by NSF of China (11071266), partially supported by Scholarship Award for Excellent Doctoral Student granted by Ministry of Education and partially supported by the found of Chongqing Normal University (13XLB006).

摘要/Abstract

摘要：

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.

Abstract:

Key words: Self-similar solutions, blow-up phenomena, two-component shallow water sys-tem

中图分类号:

35G25

周寿明, 穆春来, 王良展. SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM[J]. 数学物理学报(英文版), 2013, 33(3): 821-829.

ZHOU Shou-Ming, MU Chun-Lai, WANG Liang-Zhan. SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM[J]. Acta mathematica scientia,Series B, 2013, 33(3): 821-829.

参考文献

[1] Camassa R, Holm D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71(11): 1661–1664

[2] Chen M, Liu S Q, Zhang Y. A two-component generalization of the Camassa-Holm equation and its solutions. Lett Math Phys, 2006, 75(1): 1–15

[3] Constantin A, Ivanov R. On an integrable two-component Camassa-Holm shallow water system. Phys Lett A, 2008, 372(48): 7129–7132

[4] Dullin H R, Gottwald G A, Holm D D. Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dynam Res, 2003, 33(2): 73–79

[5] Dullin H R, Gottwald G A, Holm D D. On asymptotically equivalent shallow water wave equations. Phys D, 2004, 190: 1–14

[6] Deng Y, Xiang J, Yang T. Blowup phenomena of solutions to Euler-Poisson equations. J Math Anal Appl, 2003, 286(1): 295–306

[7] Escher J, Kohlmann M, Lenells J. The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations. J Geom Phys, 2011, 61(2): 436–452

[8] Falqui G. On a Camassa-Holm type equation with two dependent variables. J Phys A, 2006, 39(2): 327–342

[9] Fu Y, Qu C Z. Well posedness and blow-up solution for a new coupled Camassa-Holm equations with peakons. J Math Phys, 2009, 50(1): 012906

[10] Fu Y, Liu Y, Qu C Z. Well-posedness and blow-up solution for a modified two component periodic Camassa-Holm system with peakons. Math Ann, 2010, 348(2): 415–448

[11] Fu Y, Qu C Z. Unique continuation and persistence properties of solutions of the 2-component degasperis-procesi equations. Acta Mathematica Scientia, 2012, 32B(2): 652–662

[12] Gui G L, Liu Y. On the global existence and wave-breaking criteria for the two-component Camassa-Holm system. J Funct Anal, 2010, 258(12): 4251–4278

[13] Guan C X, Yin Z Y. Global existence and blow-up phenomena for an integrable twocomponent Camassa-Holm shallow water system. J Diff Equ, 2010, 248(8): 2003–2014

[14] Ivanov R. Water waves and integrability. Philos Trans Roy Soc Lond A, 2007, 365: 2267–2280

[15] Johnson R S. Camassa-Holm, Korteweg-de Vries and related models for water waves. J Fluid Mech, 2002, 455: 63–82

[16] Jin L B, Guo Z G. On a two-component Degasperis-Procesi shallow water system. Nonl Anal RWA, 2010, 11(5): 4164–4173

[17] Lai S Y, Wu Y H. Global solutions and blow-up phenomena to a shallow water equation. J Diff Equ, 2010, 249(3): 693–706

[18] Lakin W D, Sanchez D A. Topics in Ordinary Differential Equations. New York: Dover Pub Inc, 1982

[19] Liu J J, Yin Z Y. On the Cauchy problem of a two-component b-family system. Nonl Anal RWA, 2011, 12(6): 3608–3620

[20] Mu C L, Zhou S M, Zeng R. Well-posedness and blow-up phenomena for a higher order shallow water equation. J Diff Equ, 2011, 251(12): 3488–3499

[21] Mustafa O G. On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system. Wave Motion, 2009, 46(6): 397–402

[22] Ni L D. The Cauchy problem for a two-component generalized -equations. Nonl Anal TMA, 2010, 73(5): 1338–1349

[23] Popowicz Z. A 2-component or N = 2 supersymmetric Camassa-Holm equation. Phys Lett A, 2006, 354: 110-114

[24] Popowicz Z. A two-component generalization of the Degasperis-Procesi equation. J Phys A, 2006, 39(44): 13717–13726

[25] Yang L E, Ji Y S, Guo B L. The relation of two-dimensional viscous Camassa-Holm equations and the navier-stokes equations. Acta Mathematica Scientia, 2009, 29B(1): 65–73

[26] Yuen M W. Self-similar blowup solutions to the 2-component Camassa-Holm equations. J Math Phys, 2010, 51(7): 093524

[27] Yuen M W. Self-similar blowup solutions to the 2-component Degasperis-Procesi shallow water. Commun Nonlinear Sci Numer Simul, 2011, 16(9): 3463–3469

[28] Yuen M W. Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars. J Math Anal Appl, 2008, 341(1): 445–456

[29] Zhang P Z, Liu Y. Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system. Int Math Res Not, 2010, 11: 1981–2021

[30] Zhou J, Tian L X, Fan X F. Soliton, kink and antikink solutions of a 2-component of the Degasperis-Procesi equation. Nonl Anal RWA, 2010, 11(4): 2529–2536

[31] Zhou SM, Mu C L. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2013, 33(4): 1713–1739

SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM

SELF-SIMILAR SOLUTIONS AND BLOW-UP PHENOMENA FOR A TWO-COMPONENT SHALLOW WATER SYSTEM

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