On the small time asymptotics of stochastic non‐Newtonian fluids

Abstract

In this paper, a small‐time large deviation principle for the stochastic non‐Newtonian fluids driven by multiplicative noise is proved. Copyright © 2016 John Wiley & Sons, Ltd.

Small time asymptotics for SPDEs with locally monotone coefficients

“…In section 4 below, we will apply Theorem 2.2 to concrete examples of SPDE models as applications. In particular, this covers the results in [60,37], where the small time LDP for stochastic 2D Navier-Stokes equation and stochastic Ladyzhenskaya model was studied respectively.…”

“…Thus, our main result is applicable to various types of SPDEs such as stochastic porous media equation, stochastic p-Laplace equation, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, by applying the abstract result to concrete models, our main result could cover the results in [60,37], where the small time LDP for stochastic 2D Navier-Stokes equation and stochastic Ladyzhenskaya model was studied respectively. Moreover, to the best of our knowledge, our small time LDP results for general quasilinear SPDEs with multiplicative noise (such as porous media equation and p-Laplace equation) seem to be new in the literature.…”

“…Recently, the small time LDP of scalar stochastic conservation laws was also studied in [61]. The reader might refer to [12,13,33,37] and references therein for further results on this subject.…”

“…The proof of the main result here mainly follows the idea in Zhang's work [62] by using exponential equivalence arguments, which is a very powerful method used by many scholars to study the small time LDP for SPDEs, see, e.g. [12,19,37,44,50,60,61]. More precisely, consider a zero drift stochastic differential equation with the same initial data (see (4) below), where the small noise and small time asymptotics problems are equivalent.…”

“…Martingale and stationary solutions for this model was established by Guo et al in [30]. Recently, the small time LDP for this model has been studied for d = 2, p ∈ (1, 5 2 ] by Lin and Sun in [37]. Consider the Gelfand triple V ⊂ H ⊂ V * , where V = u ∈ W 2,2 0 (Λ; R d ) : div(u) = 0 in Λ ; H = u ∈ L 2 (Λ; R d ) : div(u) = 0 in Λ, u • n = 0 on ∂Λ .…”