Global existence of a stochastic shallow water wave equation under the sign condition

Youyuan Sun; Yafeng Li; Shaoyong Lai; Youyuan Sun; Yafeng Li; Shaoyong Lai

doi:10.3934/era.2026250

Electronic Research Archive

2026, Volume 34, Issue 8: 5612-5625. doi: 10.3934/era.2026250

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Global existence of a stochastic shallow water wave equation under the sign condition

1.
School of Mathematics and Statistics, Institute of Applied Mathematics, Yili Normal University, Yining 835000, China
2.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received: 16 April 2026 Revised: 17 June 2026 Accepted: 26 June 2026 Published: 02 July 2026

We investigate a stochastic shallow water wave equation in Itô form with its deterministic counterpart containing the Degasperis-Procesi and Camassa-Holm equations. The existence of a short time solution is established by transforming the equation into a stochastic transport equation endowed with the initial data in the space $ H^s(\mathbb{R}) $ $ (s > 3/2) $. Moreover, assuming that the initial value meets with the sign condition and belongs to $ H^s(\mathbb{R})\cap L^1(\mathbb{R}) $ $ (s > 3/2), $ we verify the global existence of the solution for the stochastic equation.
- stochastic shallow water wave equation,
- transport equation,
- local well-posedness,
- global well-posedness
Citation: Youyuan Sun, Yafeng Li, Shaoyong Lai. Global existence of a stochastic shallow water wave equation under the sign condition[J]. Electronic Research Archive, 2026, 34(8): 5612-5625. doi: 10.3934/era.2026250

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Abstract

We investigate a stochastic shallow water wave equation in Itô form with its deterministic counterpart containing the Degasperis-Procesi and Camassa-Holm equations. The existence of a short time solution is established by transforming the equation into a stochastic transport equation endowed with the initial data in the space $ H^s(\mathbb{R}) $ $ (s > 3/2) $. Moreover, assuming that the initial value meets with the sign condition and belongs to $ H^s(\mathbb{R})\cap L^1(\mathbb{R}) $ $ (s > 3/2), $ we verify the global existence of the solution for the stochastic equation.

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