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Cross-diffusion systems with non-zero flux and moving boundary conditions. (English) Zbl 1408.65051

Summary: We propose and analyze a one-dimensional multi-species cross-diffusion system with non-zero-flux boundary conditions on a moving domain, motivated by the modeling of a physical vapor deposition process. Using the boundedness by entropy method introduced and developped in [M. Burger et al., SIAM J. Math. Anal. 42, No. 6, 2842–2871 (2010; Zbl 1227.35155)] and [A. Jüngel, Nonlinearity 28, No. 6, 1963–2001 (2015; Zbl 1326.35175)], we prove the existence of a global weak solution to the obtained system. In addition, existence of a solution to an optimization problem defined on the fluxes is established under the assumption that the solution to the considered cross-diffusion system is unique. Lastly, we prove that in the case when the imposed external fluxes are constant and positive and the entropy density is defined as a classical logarithmic entropy, the concentrations of the different species converge in the long-time limit to constant profiles at a rate inversely proportional to time. These theoretical results are illustrated by numerical tests.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35B40 Asymptotic behavior of solutions to PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
35D35 Strong solutions to PDEs
76A20 Thin fluid films
35R37 Moving boundary problems for PDEs
82D37 Statistical mechanics of semiconductors
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[1] N.D. Alikakos, Lp bounds of solutions of reaction-diffusion equations. Commun. Partial Differ. Equ.4 (1979) 827-868. · Zbl 0421.35009 · doi:10.1080/03605307908820113
[2] H. Amann et al., Dynamic theory of quasilinear parabolic equations. ii. reaction-diffusion systems. Differ. Integral Eqs.3 (1990) 13-75. · Zbl 0729.35062
[3] A. Bakhta, Mathematical models and numerical simulation of photovoltaic devices. Ph.D. thesis, in preparation (2017).
[4] L. Boudin, B. Grec and F. Salvarani, A mathematical and numerical analysis of the maxwell-stefan diffusion equations. Discrete and Continuous Dynamical Systems-Series B17 (2012) 1427-1440. · Zbl 1245.35091 · doi:10.3934/dcdsb.2012.17.1427
[5] M. Burger, M.Di Francesco, J.-F. Pietschmann and Bärbel Schlake, Nonlinear cross-diffusion with size exclusion. SIAM J. Math. Anal.42 (2010) 2842-2871. · Zbl 1227.35155 · doi:10.1137/100783674
[6] L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion. SIAM J. Math. Anal.3 (2004) 301-322. · Zbl 1082.35075 · doi:10.1137/S0036141003427798
[7] L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Differ. Equ.224 (2006) 39-59. · Zbl 1096.35060 · doi:10.1016/j.jde.2005.08.002
[8] M. Di Francesco and J. Rosado, Fully parabolic keller-segel model for chemotaxis with prevention of overcrowding. Nonlinearity21 (2008) 2715. · Zbl 1157.35398 · doi:10.1088/0951-7715/21/11/012
[9] J. Dolbeault, B. Nazaret and G. Savaré, A new class of transport distances between measures. Calcul. Variat. Partial Differ. Equ.34 (2009) 93-231. · Zbl 1157.49042
[10] M. Dreher and A. Jüngel, Compact families of piecewise constant functions in lp (0, t; b). Nonl. Anal.: Theory, Methods Appl.75 (2012) 3072-3077. · Zbl 1245.46017 · doi:10.1016/j.na.2011.12.004
[11] J.A. Griepentrog and L. Recke. Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems. J. Evol. Equ.10 (2010) 341-375. · Zbl 1239.35084 · doi:10.1007/s00028-010-0052-4
[12] Th. Hillen and K.J. Painter, A user’s guide to pde models for chemotaxis. J. Math. Biology58 (2009) 183-217. · Zbl 1161.92003 · doi:10.1007/s00285-008-0201-3
[13] C. Lemaréchal and J.Frédréric Bonnans, J. Charles Gilbert and C. Sagastizábal, Numer. Optimiz. Theoretical Practical Aspects volume 1. Springer Verlag Berlin Heidelberg (2006).
[14] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation. SIAM J. Math. Anal.29 (1998) 1-17. · Zbl 0915.35120 · doi:10.1137/S0036141096303359
[15] A. Juengel and I. Viktoria Stelzer, Entropy structure of a cross-diffusion tumor-growth model. Math. Models Methods Appl. Sci.22 (2012) 1250009. · Zbl 1241.35119 · doi:10.1142/S0218202512500091
[16] A. Jüngel, The boundedness-by-entropy method for cross-diffusion systems. Nonl.28 (2015) 1963. · Zbl 1326.35175 · doi:10.1088/0951-7715/28/6/1963
[17] A. Jungel and I.V. Stelzer, Existence analysis of maxwell-stefan systems for multicomponent mixtures. SIAM J. Math. Anal.45 (2013) 2421-2440. · Zbl 1276.35104 · doi:10.1137/120898164
[18] K. Horst Wilhelm Küfner, Invariant regions for quasilinear reaction-diffusion systems and applications to a two population model. Nonl. Differ. Equ. Appl. NoDEA3 (1996) 421-444. · Zbl 0869.35017 · doi:10.1007/BF01193829
[19] O.A. Ladyzenskaja and V.A. Solonnikov, Nn ural ceva, linear and quasilinear equations of parabolic type, translated from the russian by s. smith. translations of mathematical monographs, vol. 23. Amer. Math. Soc., Providence, RI63 (1967) 64.
[20] D. Le and T.T. Nguyen, Everywhere regularity of solutions to a class of strongly coupled degenerate parabolic systems. Commun. Partial Differ. Equ.31 (2006) 307-324. · Zbl 1096.35030 · doi:10.1080/0360530050036038
[21] Th. Lepoutre, M. Pierre and G. Rolland, Global well-posedness of a conservative relaxed cross diffusion system. SIAM J. Math. Anal.44 (2012) 1674-1693. · Zbl 1255.35134 · doi:10.1137/110848839
[22] M. Liero and A. Mielke, Gradient structures and geodesic convexity for reaction-diffusion systems. Phil. Trans. R. Soc. A371 (2013) 20120346. · Zbl 1292.35149 · doi:10.1098/rsta.2012.0346
[23] J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, volume 1. Springer Science and Business Media (2012). · Zbl 0251.35001
[24] P. Markowich, A. Unterreiter, A. Arnold and G. Toscani, On Generalized Csiszár-Kullback Inequalities. Monatshefte für Math.131 (2000) 235-253. · Zbl 1015.94003 · doi:10.1007/s006050070013
[25] D.M. Mattox, Handbook of physical vapor deposition (PVD) processing. William Andrew (2010).
[26] K.J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis. Bulletin Math. Biology71 (2009) 1117-1147. · Zbl 1168.92010 · doi:10.1007/s11538-009-9396-8
[27] J.W. Portegies and M.A. Peletier, Well-posedness of a parabolic moving-boundary problem in the setting of wasserstein gradient flows. Preprint arXiv: 0812.1269 (2008). · Zbl 1200.35340
[28] R. Redlinger, Invariant sets for strongly coupled reaction-diffusion systems under general boundary conditions. Archive for Rational Mech. Anal.108 (1989) 281-291. · Zbl 0696.35084 · doi:10.1007/BF01052975
[29] J. Stará and O. John, Some (new) counterexamples of parabolic systems. Commentationes Mathematicae Universitatis Carolinae36 (1995) 503-510. · Zbl 0846.35024
[30] N. Zamponi and A. Jüngel, Analysis of degenerate cross-diffusion population models with volume filling. Ann. Institut Henri Poincaré (C) Non Linear Anal.34 (2017) 1-29. · Zbl 1386.35167 · doi:10.1016/j.anihpc.2015.08.003
[31] N. Zamponi and A. Jüngel, Analysis of degenerate cross-diffusion population models with volume filling (Corrigendum). Ann. Institut Henri Poincaré (C) Non Linear Analysis. 34 (2017) 789-792. · Zbl 1417.35062 · doi:10.1016/j.anihpc.2016.06.001
[32] J. Zinsl and D. Matthes, Transport distances and geodesic convexity for systems of degenerate diffusion equations. Calc. Variat. Part. Differ. Equ.54 (2015) 3397-3438. · Zbl 1342.49067 · doi:10.1007/s00526-015-0909-z
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