| On the Blow-up Phenomena of Cauchy Problem for the Camassa-Holm Equation |
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| 作者姓名： | LIU Yongqin WANG Weike |
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| 作者单位： | [1]School of Mathematics and Statistics, Wuhan Universily, Wuhan 430072, Hubei, China [2]Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, China |
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| 基金项目： | 中国科学院资助项目;上海市科委资助项目 |
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| 摘 要： | 0 IntroductionInthis paper we consider the development of singularitiesfor strong solutionto the Cauchy problemut-utxx+3uux=2uxuxx+uuxxx,t >0, x∈Ru(0, x) =u0(x) , x∈R(1) This equation was physically derived in Refs .1 ,2] todescribe the motion of solitary waves onshallow water ,whereu(t, x)describe thefree surface of the water above aflat bot-tom(or equivalently,thefluid velocityat ti metinthexdirec-tionin appropriate non-di mensional units ) .Equation (1) wasalso obtained and proved to b…
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| 关 键 词： | 初值问题 Camassa-Holm方程 单相交 偏微分方程 |
| 文章编号： | 1007-1202（2006）03-0451-05 |
| 收稿时间： | 2005-07-20 |
On the blow-up phenomena of Cauchy problem for the Camassa-Holm equation |
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| LIU Yongqin WANG Weike.On the Blow-up Phenomena of Cauchy Problem for the Camassa-Holm Equation[J].Wuhan University Journal of Natural Sciences,2006,11(3):451-455. |
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| Authors: | Liu Yongqin Wang Weike |
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| Institution: | (1) School of Mathematics and Statistics, Wuhan University, 430072 Wuhan Hubei, China;(2) Department of Mathematics, Shanghai Jiaotong University, 200030 Shanghai, China |
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| Abstract: | We focus on the blow-up phenomena of Cauchy problem for the Camassa-Holm equation. Blow-up can occur only in the form of wave-breaking, i.e. the solution is bounded but its slope becomes unbounded in finite time. We proved that there is such a point that its slope becomes infinite exactly at breaking time. We also gave the precise blow-up rate and the blow-up set. |
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| Keywords: | wave-breaking blow-up rate blow-up set |
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