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

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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.

Keywords

stochastic shallow water wave equation / transport equation / local well-posedness / global well-posedness

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