Uniqueness and generic regularity of global weak conservative solutions to the Constantin-Lannes equation

ABSTRACT

This paper is devoted to the uniqueness of global-in-time conservative solutions and generic regularity for the shallow water waves of moderate amplitude equation (Constantin-Lannes equation). The Constantin-Lannes equation possible development of singularities in finite time, and beyond the occurrence of wave breaking, it exists global conservative solutions. In the present paper, we will prove the uniqueness of global-in-time conservative solutions for the Constantin-Lannes equation with general initial data $u_0\in H^1(\mathbb{R})$ by analyzing the evolution of the quantities u and $v=2\arctan u_x$ along each characteristic. Moreover, we consider that piecewise smooth solutions with only generic singularities are dense in the whole solution set by Thom's transversality Lemma.

KEYWORDS:

• Constantin-Lannes equation
• global conservative solutions
• uniqueness
• generic regularity

MATHEMATICS SUBJECT CLASSIFICATIONS (2000):

• 65M06
• 65M12
• 35B10
• 35Q53

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author is partially supported by the National Natural Science Foundation of China [grant number 11971082], Natural Science Foundation of Chongqing [grant number csts2020jcyj-jqX0022] and the Chongqing's youth talent support program [grant number cstc2021ycjh-bgzxm0130], Science and Technology Research Program of Chongqing Municipal Educational Commission [grant number KJZD-M201900501].