Anticipating stochastic equation of two-dimensional second grade fluids

Abstract

In this paper, we consider a stochastic model of incompressible second grade fluids on a bounded domain of R 2 driven by linear multiplicative Brownian noise with anticipating initial conditions. The existence and uniqueness of the solutions are established.

Uniqueness for optimal control problems of two-dimensional second grade fluids

“…Article [7] established the well-posedness for the second grade fluid equations under a Navier slip boundary condition. Referring to the stochastic framework, the existence and uniqueness results have been investigated in [26,27,12,29,30] under non-slip and slip boundary conditions.…”

Random dynamics of two-dimensional stochastic second grade fluids

“…[15,16]. Secondly, from the anticipating/nonadapted stochastic analysis point of view, if the initial value of (1.1) is a random variable ξ(not deterministic) and measurable with respect to the σ-algebra σ(W (t) : 0 ≤ t ≤ T ) for some T > 0, then the substitute process u(t, ξ(ω), ω) will still be a solution of the anticipating SPDE (1.1) with the anticipating initial value u(0) = ξ, the proof of such a substitution result depends on the Fréchet differentiability of the solution u(t, f, ω) to (1.1), see [19]. Since the curl-term in (1.1) is order of three, the system (1.1) is highly nonlinear(far from semi-linear).…”