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multiscale; stochastic; homogenization; convection-diffusion
Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ${\partial u^\omega _{\varepsilon }}/{\partial t} +{1}/{\epsilon _3}\,\mathcal C\bigl (T_3({x}/{\varepsilon _3}) \omega _3\bigr )\cdot \nabla u^\omega _{\varepsilon }- \div \bigl ( \alpha \bigl (T_1({x}/{\varepsilon _1})\omega _1, T_2({x}/{\varepsilon _2})\omega _2 ,t\bigr ) \nabla u^\omega _{\varepsilon }\bigr )=f$. It is shown, under certain structure assumptions on the random vector field ${\mathcal C}(\omega _3)$ and the random map $\alpha (\omega _1,\omega _2,t)$, that the sequence $\lbrace u^\omega _\epsilon \rbrace $ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem ${\partial u}/{\partial t} - \div ( \mathcal B(t)\nabla u ) = f$.
[1] A.  Bensoussan, J.-L.  Lions, G.  Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam-New York-Oxford, 1978. MR 0503330
[2] V.  Chiadò Piat, G.  Dal Maso, A.  Defranceschi: G-convergence of monotone operators. Ann. Inst. H.  Poincaré, Anal. Non Linéare 7 (1990), 123–160. MR 1065871
[3] V.  Chiadò Piat, A.  Defranceschi: Homogenization of monotone operators. Nonlinear Anal., Theory Methods Appl. 14 (1990), 717–732. DOI 10.1016/0362-546X(90)90102-M | MR 1049117
[4] Y.  Efendiev, A.  Pankov: Numerical homogenization of nonlinear random parabolic operators. Multiscale Model. Simul. 2 (2004), 237–268. DOI 10.1137/030600266 | MR 2043587
[5] L. C.  Evans: Partial Differential Equations. AMS Graduate Studies in Mathematics, Vol.  19. AMS, Providence, 1998. MR 1625845
[6] A.  Fannjiang, G.  Papanicolaou: Convection enhanced diffusion for periodic flows. SIAM J.  Appl. Math. 54 (1994), 333–408. DOI 10.1137/S0036139992236785 | MR 1265233
[7] J.-L.  Lions, D.  Lukkassen, L-E.  Persson, P.  Wall: Reiterated homogenization of nonlinear monotone operators. Chin. Ann. Math., Ser.  B 22 (2001), 1–12. DOI 10.1142/S0252959901000024 | MR 1823125
[8] S.  Spagnolo: Convergence of parabolic equations. Boll. Unione Math. Ital. 14-B (1977), 547–568. MR 0460889 | Zbl 0356.35042
[9] N.  Svanstedt: G-convergence and homogenization of sequences of linear and monlinear partial differential operators. PhD. Thesis, Luleå University, 1992.
[10] N.  Svanstedt: G-convergence of parabolic operators. Nonlinear Anal., Theory Methods Appl. 36 (1999), 807–842. DOI 10.1016/S0362-546X(97)00532-4 | MR 1682689 | Zbl 0933.35020
[11] N.  Svanstedt: Multiscale stochastic homogenization of monotone operators. Netw. Heterog. Media 2 (2007), 181–192. DOI 10.3934/nhm.2007.2.181 | MR 2291817
[12] E.  Zeidler: Nonlinear Functional Analysis and its Applications  2 B. Springer, Berlin-New York, 1985. MR 0768749
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