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- W4309181505 abstract "We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa--Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $H^m$ ($minmathbb{N}$) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that balance the martingale part of the equation against the second-order Stratonovich-to-It^{o} correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional It^{o} formula and a DiPerna--Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators." @default.
- W4309181505 created "2022-11-24" @default.
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- W4309181505 date "2023-01-01" @default.
- W4309181505 modified "2023-09-25" @default.
- W4309181505 title "Global well-posedness of the viscous Camassa–Holm equation with gradient noise" @default.
- W4309181505 doi "https://doi.org/10.3934/dcds.2022163" @default.
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