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Random attractor for the Ladyzhenskaya model with additive noise. (English) Zbl 1185.35347

This paper deals with investigation of the dynamical behavior of the following \(2d\) non-Newtonian fluid with additive noise \(du=\{-(u\cdot\nabla)u-\nabla p+\nabla\cdot[2\mu_0(\varepsilon+| e|^2)^{-\alpha/2}-2\mu_1\Delta e]+g\}\,dt+\sum_{j=1}^{m}\varphi_{j}dw_{j}(t)\), \(x\in D\subset\mathbb R^2\), \(\nabla\cdot u=0\), where \(e_{ij}(u)={1\over2}({\partial u_{i}\over\partial x_{j}}+{\partial u_{j}\over\partial x_{i}})\), \(| e|^2=\sum_{i,j=1}^{n}| e_{ij}|^2\); \(\alpha\in(0,1)\); \(D\) is a bounded smooth domain of \(\mathbb R^2\); \(w_{j}(t)\;(1\leq j\leq m)\) are mutually independent two-sided Wiener processes; \(u=u(t,x,\omega)\) is a random velocity field of the fluid; \(\varphi_{j}(x), \;(1\leq j\leq m)\) are given \(2d\) vector functions. The authors use the theory of random attractor to prove the existence of a compact random attractor for the random dynamical system associated to considered model. Let us denote \({\mathcal V}=\{\varphi=(\varphi_1,\varphi_2)\in(C_{0}^{\infty}(\overline D))^2, \nabla\cdot\varphi=0\) in \(D, \varphi=0\) on \(\partial D\}\); \(H\) is the closure of \({\mathcal V}\) in \((L^2(D))^2\); \(V\) is the closure of \({\mathcal V}\) in \((H^2(D))^2\). The authors use the Ornstein-Uhlenbeck transformation to obtain some estimates of the solution in functional spaces \(H\) and \(V\), respectively, and then use the compact embedding of Sobolev spaces to obtain the existence of compact random set which absorbs any bounded nonrandom subset of space \(H\).

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35Q35 PDEs in connection with fluid mechanics
35B41 Attractors
76A05 Non-Newtonian fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
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