In this paper, we establish the well-posedness of the Fokker-Planck equation governing the anomalous dynamics of a polymer center of mass in non-static media, which exhibits a one-way coupling system of Brownian non-Gaussian diffusion equation and incompressible Navier-Stokes equations (INSE). Firstly, a stochastic representation of the solution is derived by the stochastic differential equation associated with a subordinated Brownian motion and a birth-death process, in which the drift term is the solution of the INSE on Rd(d≥2). Regarding the existence of the birth-death semigroup in ℓp(π) (p ≥ 2), equipped with a positive weight measure π, it is proved via a perturbation theorem without requiring the boundedness of the birth and death rates. Then, relying on the regularity estimates of the solution for INSE and the properties of the birth-death semigroup, we prove that the stochastic representation is a unique local strong solution in ℓp′(π)×Lp(Rd)(p′≥2,pd) via spatial regularity analyses of the random flow. The Picard iteration is used to prove that it is both unique local mild and weak solution in ℓ2(π)×L2(Rd).
Wu et al. (Wed,) studied this question.