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arXiv · 2211.07046

Global existence of dissipative solutions to the Camassa--Holm equation with transport noise

Luca Galimberti·Helge Holden·Kenneth H. Karlsen·Peter H. C. Pang
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Abstract

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--It\^{o} correction term.

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Luca Galimberti, Helge Holden, Kenneth H. Karlsen, Peter H. C. Pang. 2022-11-13. Global existence of dissipative solutions to the Camassa--Holm equation with transport noise. https://doi.org/10.1016/j.jde.2023.12.021

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