Global conservative solutions for a modified periodic coupled Camassa-Holm system

Rong Chen; Shihang Pan; Baoshuai Zhang; Rong Chen; Shihang Pan; Baoshuai Zhang

doi:10.3934/era.2020087

Electronic Research Archive

2021, Volume 29, Issue 1: 1691-1708. doi: 10.3934/era.2020087

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Global conservative solutions for a modified periodic coupled Camassa-Holm system

Rong Chen ^{1
,},
Shihang Pan ^{2
,
,},
Baoshuai Zhang ^{3
,
,}

1.
Personnel Department, Chongqing Normal University, Chongqing 401331, China
2.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3.
School of Economic Management, Chongqing Normal University, Chongqing 401331, China

Received: 01 April 2020 Revised: 01 July 2020 Published: 24 August 2020
Primary: 35A01, 35A02; Secondary: 35D30, 35G25, 35G25

In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic Coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.
- Periodic coupled Camassa-Holm system,
- global conservative solution
Citation: Rong Chen, Shihang Pan, Baoshuai Zhang. Global conservative solutions for a modified periodic coupled Camassa-Holm system[J]. Electronic Research Archive, 2021, 29(1): 1691-1708. doi: 10.3934/era.2020087

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Abstract

In present paper, we deal with the behavior of a solution beyond the occurrence of wave breaking for a modified periodic Coupled Camassa-Holm system. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, this evolution system is rewritten as a closed semilinear system. The local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows us to continue the solution after wave breaking, and gives a global conservative solution where the energy is conserved for almost all times. Returning to the original variables. We finally obtain a semigroup of global conservative solutions, which depend continuously on the initial data. Additionally, our results repair some gaps in the pervious work.

References

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