This work concerns the fluctuating hydrodynamics of various materials immersed in a fluid. Physically speaking, we work in a regime where macro- and microscopic considerations both have an effect: the particles are large enough to recognize they are in a fluid, but are small enough to feel the coarseness of the fluid's molecular nature. Pollen grains have famously been cited for exhibiting such behavior; more recently such dynamics have been noted in the study of polymers in dilute solution.
In this manuscript we develop an existence and uniqueness result for a coupled set of polymer and fluid equations. We model the fluid velocity by the three-dimensional, linearized, incompressible, stochastic Navier-Stokes equations and model the polymer by its Langevin dynamics. Similar to results that emerge from studying the stochastic heat equation, we must use correlations in the spatial structure of the noise in order to get sufficiently regular – in this case function-valued – solutions.
Along the way, we also articulate a version of the Walsh-Dalang two-parameter stochastic integral that is particularly suited to the linear SPDE theory. By classifying the driving noise according to the decay rate of its spectral measure, we connect stronger spatial correlation in the noise to increased spatial regularity for the integral process. In particular, we find conditions sufficient for a local Lipschitz property in the fluid velocity field, which enables existence and uniqueness of a passive tracer particle process driven by coloured noise.