## Multiscale stochastic homogenization of convection-diffusion equations.(English)Zbl 1199.35017

Summary: Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ${\partial u^\omega _{\varepsilon }}/{\partial t} +{1}/{\varepsilon _3}\, \mathcal C\bigl (T_3({x}/{\varepsilon _3}) \omega _3\bigr ) \cdot \nabla u^\omega _{\varepsilon }- \operatorname {div} \bigl ( \alpha \bigl (T_1({x}/{\varepsilon _1})\omega _1, T_2({x}/{\varepsilon _2})\omega _2 ,t\bigr ) \nabla u^\omega _{\varepsilon }\bigr )=f.$ 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 $$\{u^\omega _\varepsilon \}$$ of solutions converges in the sense of G-convergence of parabolic operators to the solution  $$u$$ of the homogenized problem $${\partial u}/{\partial t} - \operatorname {div} ( \mathcal B(t)\nabla u ) = f$$.

### MSC:

 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

multiscale; stochastic; homogenization; convection-diffusion
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### References:

  A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam-New York-Oxford, 1978. · Zbl 0404.35001  V. Chiadò Piat, G. Dal Maso, A. Defranceschi: G-convergence of monotone operators. Ann. Inst. H. Poincaré, Anal. Non Linéare 7 (1990), 123–160. · Zbl 0731.35033  V. Chiadò Piat, A. Defranceschi: Homogenization of monotone operators. Nonlinear Anal., Theory Methods Appl. 14 (1990), 717–732. · Zbl 0705.35041  Y. Efendiev, A. Pankov: Numerical homogenization of nonlinear random parabolic operators. Multiscale Model. Simul. 2 (2004), 237–268. · Zbl 1181.76113  L. C. Evans: Partial Differential Equations. AMS Graduate Studies in Mathematics, Vol. 19. AMS, Providence, 1998.  A. Fannjiang, G. Papanicolaou: Convection enhanced diffusion for periodic flows. SIAM J. Appl. Math. 54 (1994), 333–408. · Zbl 0796.76084  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. · Zbl 0979.35047  S. Spagnolo: Convergence of parabolic equations. Boll. Unione Math. Ital. 14-B (1977), 547–568. · Zbl 0356.35042  N. Svanstedt: G-convergence and homogenization of sequences of linear and monlinear partial differential operators. PhD. Thesis. Luleå University, 1992.  N. Svanstedt: G-convergence of parabolic operators. Nonlinear Anal., Theory Methods Appl. 36 (1999), 807–842. · Zbl 0933.35020  N. Svanstedt: Multiscale stochastic homogenization of monotone operators. Netw. Heterog. Media 2 (2007), 181–192. · Zbl 1140.35341  E. Zeidler: Nonlinear Functional Analysis and its Applications 2B. Springer, Berlin-New York, 1985. · Zbl 0583.47051
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