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 & 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.  doi: 10.1016/j.electacta.2010.02.087.  Google Scholar

[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.  doi: 10.1016/j.jcp.2012.06.005.  Google Scholar

[3]

F. Brezzi, L. D. Marini and P. Pietra, Numerical simulation of semiconductor devices,, Comput. Methods Appl. Mech. Engrg., 75 (1989), 493.  doi: 10.1016/0045-7825(89)90044-3.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1051/m2an:2003028.  Google Scholar

[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.  doi: 10.1093/imanum/23.1.81.  Google Scholar

[7]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Plenum Press, (1984).  doi: 10.1007/978-1-4757-5595-4.  Google Scholar

[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.   Google Scholar

[9]

H. B. Da Veiga, On the semiconductor drift-diffusion equations,, Differ. Int. Eqs., 9 (1996), 729.   Google Scholar

[10]

M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0,T;B)$,, Nonlinear Anal., 75 (2012), 3072.  doi: 10.1016/j.na.2011.12.004.  Google Scholar

[11]

W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations,, J. Differential Equations, 123 (1995), 523.   Google Scholar

[12]

H. Gajewski, On existence, uniqueness and asymptotic behaviour of solutions of the basic equations for carrier transport in semiconductors,, ZAMM, 65 (1985), 101.   Google Scholar

[13]

H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductors devices,, M3AS, 4 (1994), 121.   Google Scholar

[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.   Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Springer Berlin, (1984).   Google Scholar

[16]

A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors,, M3AS, 4 (1994), 677.   Google Scholar

[17]

A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in electrophoretic and semiconductor modeling,, Math. Nachr., 185 (1997), 85.   Google Scholar

[18]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001).  doi: 10.1007/978-3-0348-8334-4.  Google Scholar

[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.  doi: 10.1016/S0294-1449(99)00101-8.  Google Scholar

[20]

A. Jüngel and Y. J. Peng, Rigorous derivation of a hierarchy of macroscopic models for semiconductors and plasmas,, International Conference on Differential Equations, (1999), 1325.   Google Scholar

[21]

A. Jüngel and I. Violet, The quasi-neutral limit in the quantum drift-diffusion equations,, Asymptotic Analysis, 53 (2007), 139.   Google Scholar

[22]

P. A. Markowich, The Stationary Semiconductor Device Equations,, Computational Microelectronics, (1986).  doi: 10.1007/978-3-7091-3678-2.  Google Scholar

[23]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[24]

R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation,, Numer. Methods Partial Differential Equations, 13 (1997), 215.   Google Scholar

[25]

D. L. Scharfetter and H. K. Gummel, Large signal analysis of a silicon read diode oscillator,, IEEE Trans. Electron Dev., 16 (1969), 64.  doi: 10.1109/T-ED.1969.16566.  Google Scholar

[26]

C. Schmeiser, A singular perturbation analysis of reverse biased $pn$-junctions,, SIAM J. Math. Anal., 21 (1990), 313.  doi: 10.1137/0521017.  Google Scholar

[27]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[28]

W. Van Roosbroeck, Theory of the flow of electrons and holes in germanium and other semiconductors,, Bell System Tech. J., 29 (1950), 560.   Google Scholar

show all references

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.  doi: 10.1016/j.electacta.2010.02.087.  Google Scholar

[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.  doi: 10.1016/j.jcp.2012.06.005.  Google Scholar

[3]

F. Brezzi, L. D. Marini and P. Pietra, Numerical simulation of semiconductor devices,, Comput. Methods Appl. Mech. Engrg., 75 (1989), 493.  doi: 10.1016/0045-7825(89)90044-3.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1051/m2an:2003028.  Google Scholar

[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.  doi: 10.1093/imanum/23.1.81.  Google Scholar

[7]

F. Chen, Introduction to Plasma Physics and Controlled Fusion,, Plenum Press, (1984).  doi: 10.1007/978-1-4757-5595-4.  Google Scholar

[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.   Google Scholar

[9]

H. B. Da Veiga, On the semiconductor drift-diffusion equations,, Differ. Int. Eqs., 9 (1996), 729.   Google Scholar

[10]

M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0,T;B)$,, Nonlinear Anal., 75 (2012), 3072.  doi: 10.1016/j.na.2011.12.004.  Google Scholar

[11]

W. Fang and K. Ito, Global solutions of the time-dependent drift-diffusion semiconductor equations,, J. Differential Equations, 123 (1995), 523.   Google Scholar

[12]

H. Gajewski, On existence, uniqueness and asymptotic behaviour of solutions of the basic equations for carrier transport in semiconductors,, ZAMM, 65 (1985), 101.   Google Scholar

[13]

H. Gajewski, On the uniqueness of solutions to the drift-diffusion model of semiconductors devices,, M3AS, 4 (1994), 121.   Google Scholar

[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.   Google Scholar

[15]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order,, Springer Berlin, (1984).   Google Scholar

[16]

A. Jüngel, On the existence and uniqueness of transient solutions of a degenerate nonlinear drift-diffusion model for semiconductors,, M3AS, 4 (1994), 677.   Google Scholar

[17]

A. Jüngel, A nonlinear drift-diffusion system with electric convection arising in electrophoretic and semiconductor modeling,, Math. Nachr., 185 (1997), 85.   Google Scholar

[18]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001).  doi: 10.1007/978-3-0348-8334-4.  Google Scholar

[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.  doi: 10.1016/S0294-1449(99)00101-8.  Google Scholar

[20]

A. Jüngel and Y. J. Peng, Rigorous derivation of a hierarchy of macroscopic models for semiconductors and plasmas,, International Conference on Differential Equations, (1999), 1325.   Google Scholar

[21]

A. Jüngel and I. Violet, The quasi-neutral limit in the quantum drift-diffusion equations,, Asymptotic Analysis, 53 (2007), 139.   Google Scholar

[22]

P. A. Markowich, The Stationary Semiconductor Device Equations,, Computational Microelectronics, (1986).  doi: 10.1007/978-3-7091-3678-2.  Google Scholar

[23]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[24]

R. Sacco and F. Saleri, Mixed finite volume methods for semiconductor device simulation,, Numer. Methods Partial Differential Equations, 13 (1997), 215.   Google Scholar

[25]

D. L. Scharfetter and H. K. Gummel, Large signal analysis of a silicon read diode oscillator,, IEEE Trans. Electron Dev., 16 (1969), 64.  doi: 10.1109/T-ED.1969.16566.  Google Scholar

[26]

C. Schmeiser, A singular perturbation analysis of reverse biased $pn$-junctions,, SIAM J. Math. Anal., 21 (1990), 313.  doi: 10.1137/0521017.  Google Scholar

[27]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar

[28]

W. Van Roosbroeck, Theory of the flow of electrons and holes in germanium and other semiconductors,, Bell System Tech. J., 29 (1950), 560.   Google Scholar

[1]

Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029

[2]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure & Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97

[3]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875

[4]

Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558

[5]

Antonio Suárez. A logistic equation with degenerate diffusion and Robin boundary conditions. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1255-1267. doi: 10.3934/cpaa.2008.7.1255

[6]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731

[7]

Elio E. Espejo, Masaki Kurokiba, Takashi Suzuki. Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2627-2644. doi: 10.3934/cpaa.2013.12.2627

[8]

Corrado Lattanzio, Pierangelo Marcati. The relaxation to the drift-diffusion system for the 3-$D$ isentropic Euler-Poisson model for semiconductors. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 449-455. doi: 10.3934/dcds.1999.5.449

[9]

H.J. Hwang, K. Kang, A. Stevens. Drift-diffusion limits of kinetic models for chemotaxis: A generalization. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 319-334. doi: 10.3934/dcdsb.2005.5.319

[10]

Dietmar Oelz, Alex Mogilner. A drift-diffusion model for molecular motor transport in anisotropic filament bundles. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4553-4567. doi: 10.3934/dcds.2016.36.4553

[11]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258

[12]

Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023

[13]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[14]

Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100

[15]

Clément Jourdana, Paola Pietra. A quantum Drift-Diffusion model and its use into a hybrid strategy for strongly confined nanostructures. Kinetic & Related Models, 2019, 12 (1) : 217-242. doi: 10.3934/krm.2019010

[16]

Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157

[17]

Muriel Boulakia, Anne-Claire Egloffe, Céline Grandmont. Stability estimates for a Robin coefficient in the two-dimensional Stokes system. Mathematical Control & Related Fields, 2013, 3 (1) : 21-49. doi: 10.3934/mcrf.2013.3.21

[18]

Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033

[19]

Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767

[20]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (2)

[Back to Top]