Uniqueness and stability of global conservative solutions for the modified coupled Camassa–Holm system

ABSTRACT

In this paper, we investigate the generic regularity of global-in-time conservative solutions for the modified coupled Camassa–Holm (CH) system on real line. It has been shown that the modified coupled CH system possible development of singularities in finite time and beyond the occurrence of wave breaking the global conservative solution is existent. In this paper, we further prove that the Cauchy problem of the modified coupled CH system with the initial data $z_0=(u_0,v_0)\in H^1\left(\mathbb{R}\right)\times H^1\left(\mathbb{R}\right)$ has a unique global conservative weak solution. We also study the stability for the unique global conservative solutions. Therefore, we obtain the generic regularity of the conservative weak solutions.

KEYWORDS:

• Modified coupled Camassa–Holm system
• global conservative weak solutions
• uniqueness
• characteristics
• generic regularity

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 35B10
• 35Q53

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Disclosure statement

No potential conflict of interest was reported by the authors.

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Funding

The second author Zhou is supported by is supported in part by the National Science Foundation of China (Grant No. 11971082 and 11771063), Science and Technology Research Program of Chongqing Municipal Educational Commission (No. KJZD-M201900501 and KJZD-M201800501). The third author Zhang is supported by the National Social Science Fund of China (19BJY077), Science and Technology Research Program of Chongqing Municipal Educational Committee, the Chongqing Normal University Fund(13XWB00, 16XYY268), Chongqing Social Science Planning Project (2018PY74), Science and Technology Project of Chongqing Education Commission (KJQN20180051).