Lipschitz metric for conservative solutions of the modified two-component Camassa–Holm system
Lipschitz metric for conservative solutions of the modified two-component Camassa–Holm system
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DOI：
10.1007/s00033-018-0992-z
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发表时间：
2017-02
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影响因子：
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通讯作者：
H. Cai;Zhong Tan
中科院分区：
文献类型：
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作者：
H. Cai;Zhong Tan
本文研究了一类修正的二分量Camassa-Holm方程组保守解的Lipschitz连续依赖性。即使对于光滑的初始数据，已知这些解的梯度可以在有限时间内爆炸。当这种情况发生时，由方程产生的流动不能是Lipschitz连续的w.r.t.这是索伯列夫空间中常见的距离为了科普这个问题，我们构造了一个新的距离，产生的Finsler结构，这使得Lipschitz连续的保守解的半群。我们的距离首先构造在充分正则解的子集上，然后通过连续性扩展到整个空间。这是可能的，这要归功于一个通用的规律性结果，它具有独立的兴趣。粗略地说，我们表明，一般光滑的初始数据，解决方案仍然分段光滑。
In this paper, we study the Lipschitz continuous dependence of conservative solutions to a modified two-component Camassa–Holm system. Even for smooth initial data, it is known that the gradient of these solutions can blow up in finite time. When this happens, the flow generated by the equations fails to be Lipschitz continuous w.r.t. the usual distance in the Sobolev space. To cope with this issue, we construct a new distance, generated by a Finsler structure, which renders Lipschitz continuous the semigroup of conservative solutions. Our distance is constructed first on a subset of sufficiently regular solutions, then extended by continuity to the entire space. This is possible thanks to a generic regularity result, which has independent interest. Roughly speaking, we show that, for generic smooth initial data, the solution remains piecewise smooth.