A local-in-time theory for singular SDEs with applications to fluid models with transport noise
arXiv:2010.09972 · doi:10.1007/s00332-021-09755-9
Abstract
In this paper, we establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in some Hilbert space. The key requirement is an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. With a cancellation estimate for generalized Lie derivative operators, we can construct such regular approximations for cases involving the Lie derivative operators, or more generally, differential operators of order one with suitable coefficients. In particular, we apply the abstract theory to achieve novel local-in-time results for the stochastic two-component Camassa--Holm (CH) system and for the stochastic Córdoba-Córdoba-Fontelos (CCF) model.
References in corpus (16)
- An integrable shallow water equation with peaked solitons
- Well-posedness of the transport equation by stochastic perturbation
- Variational Principles for Stochastic Fluid Dynamics
- Local Martingale and Pathwise Solutions for an Abstract Fluids Model
- Noise Prevents Singularities in Linear Transport Equations
- Solution properties of a 3D stochastic Euler fluid equation
- Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics
- Wave breaking for the Stochastic Camassa-Holm equation
- Incompressible flow in porous media with fractional diffusion
- On the well-posedness of stochastic Boussinesq equations with cylindrical multiplicative noise
- Modelling the climate and weather of a 2D Lagrangian-averaged Euler-Boussinesq equation with transport noise
- Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations
- Lagrangian averaged stochastic advection by Lie transport for fluids
- Stochastic Integral with respect to Cylindrical Wiener Process
- A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation
- On a stochastic Camassa-Holm type equation with higher order nonlinearities