Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion
Oscillation-induced blow-up to the modified Camassa–Holm equation with linear dispersion
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DOI：
10.1016/j.aim.2014.12.003
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发表时间：
2015-02
影响因子：
1.7
通讯作者：
R. Chen;Yue Liu;C. Qu;Shuanghu Zhang
中科院分区：
文献类型：
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作者：
R. Chen;Yue Liu;C. Qu;Shuanghu Zhang
本文给出了具有变线性色散的修正Camassa-Holm方程的爆破机制。我们首先考虑的情况下，线性色散是不存在的，并得出一个有限时间爆破的结果与初始数据具有温和的振荡区域。分析的一个关键特征是发展了Burgers型不等式，该不等式具有聚焦于特征的性质，可以通过跟踪解与其梯度之间的比率来推导。利用解的连续性和单调性，将该爆破准则推广到负线性色散的情形，并确定了当初始动量密度小于线性色散的大小且初始数据具有局部的弱振荡区域时，有限时间爆破仍然可以发生.最后，我们证明了在非负线性色散的情况下，奇点的形成可以由具有足够陡峭的剖面的初始基准面引起。与具有线性色散的Camassa-Holm方程相比，修正的Camassa-Holm方程的线性色散对爆破现象的影响是相当微妙的。
In this paper, we provide a blow-up mechanism to the modified Camassa–Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics, which can be deduced from tracing the ratio between solution and its gradient. Using the continuity and monotonicity of the solutions, we then extend this blow-up criterion to the case of negative linear dispersion, and determine that the finite time blow-up can still occur if the initial momentum density is bounded below by the magnitude of the linear dispersion and the initial datum has a local mild-oscillation region. Finally, we demonstrate that in the case of non-negative linear dispersion the formation of singularities can be induced by an initial datum with a sufficiently steep profile. In contrast to the Camassa–Holm equation with linear dispersion, the effect of linear dispersion of the modified Camassa–Holm equation on the blow-up phenomena is rather delicate.