一个n分支 μ-Camassa-Holm系统解的局部适定性和爆破现象研究
Local Well-Posedness and Blow-Up Phenomena of Solutions of a n-Component μ-Camassa-Holm System
DOI: 10.12677/aam.2024.138372, PDF,    科研立项经费支持
作者: 高亚琴, 王海权*, 滕凯民：太原理工大学数学学院，山西 太原
关键词: n分支-Camassa-Holm系统；局部适定性；爆破现象；A n-Component -Camassa-Holm System； Local Well-Posedness； Blow-Up Phenomena
摘要: 首先利用Kato理论，研究了一个具有多尖峰孤子解和满足 H 1 守恒的n分支 μ -Camassa-Holm系统Cauchy问题解的局部适定性；然后利用守恒律和能量估计，研究了该系统解的爆破现象。
Abstract: By utilizing Kato theory, this paper first establishes the local well-posedness of the solutions of the Cauchy problem of a n-component μ -Camassa-Holm system with multi-peakons and H 1 -conservation law. Then, the blow-up phenomena of the solutions is studied by means of conservation law and energy estimations.
文章引用：高亚琴, 王海权, 滕凯民. 一个n分支 μ-Camassa-Holm系统解的局部适定性和爆破现象研究[J]. 应用数学进展, 2024, 13(8): 3903-3916. https://doi.org/10.12677/aam.2024.138372

参考文献

[1] 李颖颖, 闫璐. 具有尖峰孤子解的三分量μ-Camassa-Holm方程组[J]. 纯粹数学与应用数学, 2020, 36(2): 168-183.
[2] Khesin, B., Lenells, J. and Misiołek, G. (2008) Generalized Hunter-Saxton Equation and the Geometry of the Group of Circle Diffeomorphisms. Mathematische Annalen, 342, 617-656. [Google Scholar] [CrossRef] 
[3] Fu, Y., Liu, Y. and Qu, C. (2012) On the Blow-Up Structure for the Generalized Periodic Camassa-Holm and Degasperis-Procesi Equations. Journal of Functional Analysis, 262, 3125-3158. [Google Scholar] [CrossRef] 
[4] Lenells, J., Misiołek, G. and Tiğlay, F. (2010) Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions. Communications in Mathematical Physics, 299, 129-161. [Google Scholar] [CrossRef] 
[5] Li, Y., Fu, Y. and Qu, C. (2020) The Two-Component-Camassa-Holm System with Peaked Solutions. Discrete & Continuous Dynamical Systems A, 40, 5929-5954. [Google Scholar] [CrossRef] 
[6] Constantin, A. (2000) On the Blow-Up of Solutions of a Periodic Shallow Water Equation. Journal of Nonlinear Science, 10, 391-399. [Google Scholar] [CrossRef] 
[7] Constantin, A. and Escher, J. (1998) Wave Breaking for Nonlinear Nonlocal Shallow Water Equations. Acta Mathematica, 181, 229-243. [Google Scholar] [CrossRef] 
[8] Constantin, A. and Escher, J. (2000) On the Blow-Up Rate and the Blow-Up Set of Breaking Waves for a Shallow Water Equation. Mathematische Zeitschrift, 233, 75-91. [Google Scholar] [CrossRef] 
[9] Wu, X. and Yin, Z. (2012) Well-Posedness and Global Existence for the Novikov Equation. Annali Scuola Normale Superiore-Classe di Scienze, 11, 707-727. [Google Scholar] [CrossRef] 
[10] Fu, Y., Gui, G., Liu, Y. and Qu, C. (2013) On the Cauchy Problem for the Integrable Modified Camassa-Holm Equation with Cubic Nonlinearity. Journal of Differential Equations, 255, 1905-1938. [Google Scholar] [CrossRef] 
[11] Gui, G., Liu, Y., Olver, P.J. and Qu, C. (2012) Wave-Breaking and Peakons for a Modified Camassa-Holm Equation. Communications in Mathematical Physics, 319, 731-759. [Google Scholar] [CrossRef] 
[12] Qu, C., Fu, Y. and Liu, Y. (2014) Well-Posedness, Wave Breaking and Peakons for a Modified Μ-Camassa-Holm Equation. Journal of Functional Analysis, 266, 433-477. [Google Scholar] [CrossRef] 
[13] Kato, T. (1975) Quasi-Linear Equations of Evolution, with Applications to Partial Differential Equations. In: Everitt, W.N., Ed., Spectral Theory and Differential Equations, Springer, Berlin, 25-70. [Google Scholar] [CrossRef] 
[14] Kato, T. and Ponce, G. (1988) Commutator Estimates and the Euler and Navier‐Stokes Equations. Communications on Pure and Applied Mathematics, 41, 891-907. [Google Scholar] [CrossRef] 
[15] Iorio Jr, R.J. and Iorio, V.d.M. (2001) Fourier Analysis and Partial Differential Equations. Cambridge University Press, Cambridge. [Google Scholar] [CrossRef] 
[16] Evans, L.C. (1998) Partial Differential Equations. Journal of Symbolic Logic, American Mathematical Society, 19, 623-624.
[17] Fu, Y. and Qu, C. (2009) Well Posedness and Blow-Up Solution for a New Coupled Camassa-Holm Equations with Peakons. Journal of Mathematical Physics, 50. [Google Scholar] [CrossRef] 

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