# American Institute of Mathematical Sciences

January  2015, 20(1): 77-92. doi: 10.3934/dcdsb.2015.20.77

## On the existence of solutions for a drift-diffusion system arising in corrosion modeling

 1 Laboratoire Paul Painlevé, CNRS UMR 8524, Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France 2 Laboratoire Paul Painlevé, CNRS-UMR 8524, Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received  March 2014 Revised  March 2014 Published  November 2014

In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time.
Citation: Claire Chainais-Hillairet, Ingrid Lacroix-Violet. On the existence of solutions for a drift-diffusion system arising in corrosion modeling. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 77-92. doi: 10.3934/dcdsb.2015.20.77
##### References:
 [1] C. Bataillon, C. Chainais-Hillairet, C. Desgranges, E. Hoarau, F. Martin, S. Perrin, M. Turpin and J. Talandier, Corrosion modelling of iron based alloy in nuclear waste repository, Electrochimica Acta, 55 (2010), 4451-4467. doi: 10.1016/j.electacta.2010.02.087. [2] C. Bataillon, F. Bouchon, C. Chainais-Hillairet, J. Fuhrmann, E. Hoarau and R. Touzani, Numerical methods for the simulation of a corrosion model with moving oxide layer, Journal of Computational Physics, 231 (2012), 6213-6231. doi: 10.1016/j.jcp.2012.06.005. [3] F. Brezzi, L. D. Marini and P. Pietra, Numerical simulation of semiconductor devices, Comput. Methods Appl. Mech. Engrg., 75 (1989), 493-514. doi: 10.1016/0045-7825(89)90044-3. [4] C. Chainais-Hillairet and I. Lacroix-Violet, Existence of solutions for a steady state corrosion of steel model, Applied Math. Letters, 25 (2012), 1784-1789. [5] C. Chainais-Hillairet, J. G. Liu and Y. J. Peng, Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis, M2AN Math. Model. Numer. Anal., 37 (2003), 319-338. doi: 10.1051/m2an:2003028. [6] C. Chainais-Hillairet and Y. J. Peng, Convergence of a finite-volume scheme for the drift-diffusion equations in 1D, IMA J. Numer. Anal., 23 (2003), 81-108. doi: 10.1093/imanum/23.1.81. [7] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum Press, New-York, 1984. doi: 10.1007/978-1-4757-5595-4. [8] Z. Chen and B. Cockburn, Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case, Numer. Math., 71 (1995), 1-28. [9] H. B. Da Veiga, On the semiconductor drift-diffusion equations, Differ. Int. Eqs., 9 (1996), 729-744. [10] M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0,T;B)$, Nonlinear Anal., 75 (2012), 3072-3077. doi: 10.1016/j.na.2011.12.004. [11] W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations, 123 (1995), 523-566. [12] H. Gajewski, On existence, uniqueness and asymptotic behaviour of solutions of the basic equations for carrier transport in semiconductors, ZAMM, 65 (1985), 101-108. [13] H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductors devices, M3AS, 4 (1994), 121-133. [14] I. Gasser, The initial time layer problem and the quasineutral limit in a nonlinear drift diffusion model for semiconductors, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 237-249. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer Berlin, New-York, 1984. [16] A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, M3AS, 4 (1994), 677-703. [17] A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in electrophoretic and semiconductor modeling, Math. Nachr., 185 (1997), 85-110. [18] A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations and their Applications, 41, Birkhäuser Verlag, 2001. doi: 10.1007/978-3-0348-8334-4. [19] A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations, Ann. Inst. H. Poincaré, 17 (2000), 83-118. doi: 10.1016/S0294-1449(99)00101-8. [20] A. Jüngel and Y. J. Peng, Rigorous derivation of a hierarchy of macroscopic models for semiconductors and plasmas, International Conference on Differential Equations, 1 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 1325-1327. [21] A. Jüngel and I. Violet, The quasi-neutral limit in the quantum drift-diffusion equations, Asymptotic Analysis, 53 (2007), 139-157. [22] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, 1986. doi: 10.1007/978-3-7091-3678-2. [23] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990. doi: 10.1007/978-3-7091-6961-2. [24] R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation, Numer. Methods Partial Differential Equations, 13 (1997), 215-236. [25] D. L. Scharfetter and H. K. Gummel, Large signal analysis of a silicon read diode oscillator, IEEE Trans. Electron Dev., 16 (1969), 64-77. doi: 10.1109/T-ED.1969.16566. [26] C. Schmeiser, A singular perturbation analysis of reverse biased $pn$-junctions, SIAM J. Math. Anal., 21 (1990), 313-326. doi: 10.1137/0521017. [27] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [28] W. Van Roosbroeck, Theory of the flow of electrons and holes in germanium and other semiconductors, Bell System Tech. J., 29 (1950), 560-607.

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##### References:
 [1] C. Bataillon, C. Chainais-Hillairet, C. Desgranges, E. Hoarau, F. Martin, S. Perrin, M. Turpin and J. Talandier, Corrosion modelling of iron based alloy in nuclear waste repository, Electrochimica Acta, 55 (2010), 4451-4467. doi: 10.1016/j.electacta.2010.02.087. [2] C. Bataillon, F. Bouchon, C. Chainais-Hillairet, J. Fuhrmann, E. Hoarau and R. Touzani, Numerical methods for the simulation of a corrosion model with moving oxide layer, Journal of Computational Physics, 231 (2012), 6213-6231. doi: 10.1016/j.jcp.2012.06.005. [3] F. Brezzi, L. D. Marini and P. Pietra, Numerical simulation of semiconductor devices, Comput. Methods Appl. Mech. Engrg., 75 (1989), 493-514. doi: 10.1016/0045-7825(89)90044-3. [4] C. Chainais-Hillairet and I. Lacroix-Violet, Existence of solutions for a steady state corrosion of steel model, Applied Math. Letters, 25 (2012), 1784-1789. [5] C. Chainais-Hillairet, J. G. Liu and Y. J. Peng, Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis, M2AN Math. Model. Numer. Anal., 37 (2003), 319-338. doi: 10.1051/m2an:2003028. [6] C. Chainais-Hillairet and Y. J. Peng, Convergence of a finite-volume scheme for the drift-diffusion equations in 1D, IMA J. Numer. Anal., 23 (2003), 81-108. doi: 10.1093/imanum/23.1.81. [7] F. Chen, Introduction to Plasma Physics and Controlled Fusion, Plenum Press, New-York, 1984. doi: 10.1007/978-1-4757-5595-4. [8] Z. Chen and B. Cockburn, Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case, Numer. Math., 71 (1995), 1-28. [9] H. B. Da Veiga, On the semiconductor drift-diffusion equations, Differ. Int. Eqs., 9 (1996), 729-744. [10] M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0,T;B)$, Nonlinear Anal., 75 (2012), 3072-3077. doi: 10.1016/j.na.2011.12.004. [11] W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations, J. Differential Equations, 123 (1995), 523-566. [12] H. Gajewski, On existence, uniqueness and asymptotic behaviour of solutions of the basic equations for carrier transport in semiconductors, ZAMM, 65 (1985), 101-108. [13] H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductors devices, M3AS, 4 (1994), 121-133. [14] I. Gasser, The initial time layer problem and the quasineutral limit in a nonlinear drift diffusion model for semiconductors, NoDEA Nonlinear Differential Equations Appl., 8 (2001), 237-249. [15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Springer Berlin, New-York, 1984. [16] A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors, M3AS, 4 (1994), 677-703. [17] A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in electrophoretic and semiconductor modeling, Math. Nachr., 185 (1997), 85-110. [18] A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations and their Applications, 41, Birkhäuser Verlag, 2001. doi: 10.1007/978-3-0348-8334-4. [19] A. Jüngel and Y. J. Peng, A hierarchy of hydrodynamic models for plasmas. Zero-electron-mass limits in the drift-diffusion equations, Ann. Inst. H. Poincaré, 17 (2000), 83-118. doi: 10.1016/S0294-1449(99)00101-8. [20] A. Jüngel and Y. J. Peng, Rigorous derivation of a hierarchy of macroscopic models for semiconductors and plasmas, International Conference on Differential Equations, 1 (Berlin, 1999), World Sci. Publ., River Edge, NJ, 2000, 1325-1327. [21] A. Jüngel and I. Violet, The quasi-neutral limit in the quantum drift-diffusion equations, Asymptotic Analysis, 53 (2007), 139-157. [22] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer-Verlag, 1986. doi: 10.1007/978-3-7091-3678-2. [23] P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990. doi: 10.1007/978-3-7091-6961-2. [24] R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation, Numer. Methods Partial Differential Equations, 13 (1997), 215-236. [25] D. L. Scharfetter and H. K. Gummel, Large signal analysis of a silicon read diode oscillator, IEEE Trans. Electron Dev., 16 (1969), 64-77. doi: 10.1109/T-ED.1969.16566. [26] C. Schmeiser, A singular perturbation analysis of reverse biased $pn$-junctions, SIAM J. Math. Anal., 21 (1990), 313-326. doi: 10.1137/0521017. [27] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. [28] W. Van Roosbroeck, Theory of the flow of electrons and holes in germanium and other semiconductors, Bell System Tech. J., 29 (1950), 560-607.
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