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On the nonlinear convectiondiffusionreaction problem in a thin domain with a weak boundary absorption
1.  Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30,10000, Zagreb, Croatia 
2.  Department of Applied Mathematics, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, SP, Brazil 
Motivated by the applications from chemical engineering, in this paper we present a rigorous derivation of the effective model describing the convectiondiffusionreaction process in a thin domain. The problem is described by a nonlinear elliptic problem with nonlinearity appearing both in the governing equation as well in the boundary condition. Using rigorous analysis in appropriate functional setting, we show that the starting singular problem posed in a twodimensional region can be approximated with one which is regular, onedimensional and captures the effects of all physical processes which are relevant for the original problem.
References:
[1] 
G. Allaire, A.L. Raphael, Homogenization of a convectiondiffusion model with reaction in a porous medium, C. R. Acad. Sci. Ser. I, 344 (2007), 523528. 
[2] 
G. S. Aragão, A. L. Pereira, M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Meth. Appl. Sci., 35 (2012), 11101116. 
[3] 
G. S. Aragão, A. L. Pereira, M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating in the boundary, J. Dyn. Differ. Equ., 26 (2014), 871888. 
[4] 
R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. London Sect. A, 235 (1956), 6777. 
[5] 
J. M. Arrieta, A. JiménezCasas, A. RodríguezBernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Revista Matemática Iberoamericana, 24 (2008), 183211. 
[6] 
J. M. Arrieta, M. C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, J. Math. Anal. Appl., 404 (2013), 86104. 
[7] 
V. Balasubramanian, G. Jayaraman and S. R. K. Iyengar, Effect of secondary flows in contaminant dispersion with weak boundary absorption Appl. Math. Model. 21 (1997), 275285. 
[8] 
S. R. M. Barros, M. C. Pereira, Semilinear elliptic equations in thin domains with reaction terms concentrating on boundary, J. Math. Anal. Appl., 441 (2016), 375392. 
[9] 
L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, 19. American Mathematical Society, 2010. 
[10] 
P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman Advanced Publishing Program, 1985. 
[11] 
J. K. Hale, G. Raugel, Reactiondiffusion equation on thin domains, J. Math. Pures et Appl., 71 (1992), 3395. 
[12] 
A. JiménezCasas, A. RodríguezBernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. Appl., 379 (2011), 567588. 
[13] 
M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis SpringerVerlag, New York, 1984. 
[14] 
O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Ellipitic Equations Academic Press, 1968. 
[15] 
E. MarušićPaloka, I. Pažanin, On the reactive solute transport through a curved pipe, Appl. Math. Lett., 24 (2011), 878882. 
[16] 
A. Mikelić, V. Devigne, C. J. van Duijn, Rigorous upscaling of the reactive flow through a pore, under dominant Péclet and Damkohler numbers, SIAM J. Math. Anal., 38 (2006), 12621287. 
[17] 
I. Pažanin, Modelling of solute dispersion in a circular pipe filled with a micropolar fluid, Math. Comp. Model., 57 (2013), 23662373. 
[18] 
M. C. Pereira, Remarks on Semilinear Parabolic Systems with terms concentrating in the boundary, Nonlinear Anal. Real World Appl., 14 (2013), 19211930. 
[19] 
P. G. Siddheshwar, S. Manjunath, Unsteady convectivediffusion with heterogeneous chemical reaction in a planePoseuille flow of a micropolar fluid, Int. J. Engng. Sci., 38 (2000), 765783. 
[20] 
G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. London Sect. A, 219 (1953), 186203. 
[21] 
G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647687. 
[22] 
H. F. Woolard, J. Billingham, O. E. Jensen, G. Lian, A multiscale model for solute transport in a wavywalled channel, J. Eng. Math., 64 (2009), 2548. 
show all references
References:
[1] 
G. Allaire, A.L. Raphael, Homogenization of a convectiondiffusion model with reaction in a porous medium, C. R. Acad. Sci. Ser. I, 344 (2007), 523528. 
[2] 
G. S. Aragão, A. L. Pereira, M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Math. Meth. Appl. Sci., 35 (2012), 11101116. 
[3] 
G. S. Aragão, A. L. Pereira, M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating in the boundary, J. Dyn. Differ. Equ., 26 (2014), 871888. 
[4] 
R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. London Sect. A, 235 (1956), 6777. 
[5] 
J. M. Arrieta, A. JiménezCasas, A. RodríguezBernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Revista Matemática Iberoamericana, 24 (2008), 183211. 
[6] 
J. M. Arrieta, M. C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, J. Math. Anal. Appl., 404 (2013), 86104. 
[7] 
V. Balasubramanian, G. Jayaraman and S. R. K. Iyengar, Effect of secondary flows in contaminant dispersion with weak boundary absorption Appl. Math. Model. 21 (1997), 275285. 
[8] 
S. R. M. Barros, M. C. Pereira, Semilinear elliptic equations in thin domains with reaction terms concentrating on boundary, J. Math. Anal. Appl., 441 (2016), 375392. 
[9] 
L. C. Evans, Partial Differential Equations Graduate Studies in Mathematics, 19. American Mathematical Society, 2010. 
[10] 
P. Grisvard, Elliptic Problems in Nonsmooth Domains Pitman Advanced Publishing Program, 1985. 
[11] 
J. K. Hale, G. Raugel, Reactiondiffusion equation on thin domains, J. Math. Pures et Appl., 71 (1992), 3395. 
[12] 
A. JiménezCasas, A. RodríguezBernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, J. Math. Anal. Appl., 379 (2011), 567588. 
[13] 
M. A. Krasnoselskii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis SpringerVerlag, New York, 1984. 
[14] 
O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Ellipitic Equations Academic Press, 1968. 
[15] 
E. MarušićPaloka, I. Pažanin, On the reactive solute transport through a curved pipe, Appl. Math. Lett., 24 (2011), 878882. 
[16] 
A. Mikelić, V. Devigne, C. J. van Duijn, Rigorous upscaling of the reactive flow through a pore, under dominant Péclet and Damkohler numbers, SIAM J. Math. Anal., 38 (2006), 12621287. 
[17] 
I. Pažanin, Modelling of solute dispersion in a circular pipe filled with a micropolar fluid, Math. Comp. Model., 57 (2013), 23662373. 
[18] 
M. C. Pereira, Remarks on Semilinear Parabolic Systems with terms concentrating in the boundary, Nonlinear Anal. Real World Appl., 14 (2013), 19211930. 
[19] 
P. G. Siddheshwar, S. Manjunath, Unsteady convectivediffusion with heterogeneous chemical reaction in a planePoseuille flow of a micropolar fluid, Int. J. Engng. Sci., 38 (2000), 765783. 
[20] 
G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. London Sect. A, 219 (1953), 186203. 
[21] 
G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Anal., 2 (1978), 647687. 
[22] 
H. F. Woolard, J. Billingham, O. E. Jensen, G. Lian, A multiscale model for solute transport in a wavywalled channel, J. Eng. Math., 64 (2009), 2548. 
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