Well-posedness of a stochastic Navier-Stokes system with dynamically coupled subgrid scales

We study a stochastic Navier–Stokes system in which the spatial modes of the transport noise evolve dynamically and are coupled to the resolved velocity. The model couples a Navier–Stokes stochastic PDE to an infinite family of linearized Navier-Stokes equations for the noise correlation modes. These latter equations may contain a hyperviscosity. We prove existence of martingale solutions in two dimensions for hyperviscosity exponent $s\ge 1$, and pathwise uniqueness for $s>1$ under additional regularity of the initial noise modes. In three dimensions, we establish existence of martingale solutions for $s>3\mathrm{/}2$. The proof combines Galerkin approximations and compactness with estimates adapted to the coupled drift and stochastic terms.