Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations

Abstract

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa--Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

References in corpus (7)

• An integrable shallow water equation with peaked solitons
• An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
• Well-posedness of the transport equation by stochastic perturbation
• Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary pde
• Noise Prevents Singularities in Linear Transport Equations
• A nonlocal shallow-water model arising from the full water waves with the Coriolis effect
• Bismut Formula for Lions Derivative of Distribution-Path Dependent SDEs

Cited by in corpus (4)

• Well-posedness for a stochastic Camassa-Holm type equation with higher order nonlinearities
• Distribution Dependent Stochastic Differential Equations
• A local-in-time theory for singular SDEs with applications to fluid models with transport noise
• Large Deviation Principle for McKean-Vlasov Quasilinear Stochastic Evolution Equations