Well-posedness at vanishing correlation time

Establish the well-posedness of the coupled stochastic water-wave system in the limit of vanishing correlation time.

Background

The paper develops coupled stochastic water-wave equations in which a large-scale potential-flow component interacts with regularized unresolved-scale fluctuations. The finite-correlation-time formulation is formally well defined, but the decorrelation limit becomes singular because quadratic nonlinearities involving gradients of the stochastic potential lack a straightforward interpretation.

The authors introduce a prescribed-noise, Hamiltonian reduction based on a small-scale Bernoulli constraint to obtain a formally meaningful zero-correlation-time model. A rigorous analysis establishing existence, uniqueness, stability, or other well-posedness properties for the resulting coupled system at vanishing correlation time is not provided and is explicitly identified as an open mathematical problem.