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Semi-classical solutions for a nonlinear coupled elliptic-parabolic problem

Published online by Cambridge University Press:  17 April 2009

Catherine Choquet
Catherine Choquet
Affiliation:
LATP, CNRS UMR 6632, Université Paul Cézanne (Aix-Marseille III), FST, Case Cour 13397 Marseille Cedex 20, France e-mail: [email protected]
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We give an existence result for a fully nonlinear system consisting of a parabolic equation strongly coupled with an elliptic one. It models in particular miscible displacement in porous media. To this aim, we adapt the tools of Ladyzenskaja, Solon-nikov and Uralćeva [27, 28] to the coupled nonlinear setting. Under some reasonable assumptions on the data, we state the existence of semi-classical solutions for the problem. We also give an existence result of weak solutions for a degenerate form of the problem.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 3 , December 2007 , pp. 369 - 396
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Alt, H.W. and DiBenedetto, E., ‘Flow of oil and water through porous media. Variational methods for equilibrium problems of fluids (Trento, 1983)’, Astérisque 118 (1984), 89–108.Google Scholar
[2]Amaziane, B., Bourgeat, A. and Jurak, M., ‘Effective macrodiffusion in solute transport through heterogeneous porous media’, Multiscale Model. Simul. 5 (2006), 184–204.CrossRefGoogle Scholar
[3]Amirat, Y. and Moussaoui, M., ‘Analysis of a one-dimensional model for compressible miscible displacement in porous media’, SIAM J. Math. Anal. 26 (1995), 659–674.CrossRefGoogle Scholar
[4]Amirat, Y. and Ziani, A., ‘Classical solutions of a parabolic-hyperbolic system modeling a three-dimensional compressible miscible flow in porous media’, Appl. Anal. 72 (1999), 155–168.CrossRefGoogle Scholar
[5]Amirat, Y. and Ziani, A., ‘Asymptotic behavior of the solutions of an elliptic-parabolic system arising in flow in porous media’, Z. Anal. Andwendungen 23 (2004), 335–351.CrossRefGoogle Scholar
[6]Anderson, M.P., ‘Movement of contaminants in groundwater: Groundwater transport - Advection and dispersion.’, in Groundwater contamination (National Academy Press, Washington, DC, 1984), pp. 37–45.Google Scholar
[7]Balasuriya, S. and Jones, C.K.R.T., ‘Diffusive draining and growth of eddies’, Nonlin. Processes Geophysics 8 (2001), 241–251.CrossRefGoogle Scholar
[8]Biferale, L., Crisanti, A., Vergassola, M. and Vulpiani, A., ‘Eddy diffusivities in scalar transport’, Phys. Fluids 7 (1995), 2725–2734.CrossRefGoogle Scholar
[9]Caffarelli, L.A. and Peral, I., ‘On W1, p estimates for elliptic equations in divergence form’, Comm. Pure Appl. Math. 51 (1998), 1–21.3.0.CO;2-G>CrossRefGoogle Scholar
[10]Chechkin, A.V., Tur, A.V. and Yanovsky, V.V., ‘Anomalous flows of passive admixture in helical turbulence’, Geophys. Astrophys. Fluid Dynam. 88 (1998), 187–213.CrossRefGoogle Scholar
[11]Chen, Z. and Ewing, R., ‘Mathematical analysis for reservoir models’, SIAM J. Math. Anal. 30 (1999), 431–453.CrossRefGoogle Scholar
[12]Chen, Z. and Khlopina, N.L., ‘Degenerate two-phase incompressible flow problems. II. Error estimates’, Commun. Appl. Anal. 5 (2001), 503–521.Google Scholar
[13]Choquet, C., ‘Existence result for a radionuclide transport model with an unbounded viscosity’, J. Math. Fluid Mech. 6 (2004), 365–388.CrossRefGoogle Scholar
[14]Choquet, C., ‘On a nonlinear parabolic system modelling miscible compressible displacement in porous media’, Nonlinear Anal. 61 (2005), 237–260.CrossRefGoogle Scholar
[15]de Marsily, G., Hydrogéologie quantitative (Masson, Paris, 1981).Google Scholar
[16]Douglas, J. and Roberts, J.E., ‘Numerical methods for a model of compressible miscible displacement in porous media’, Math. Comp. 41 (1983), 441–459.CrossRefGoogle Scholar
[17]Fabrie, P. and Langlais, M., ‘Mathematical analysis of miscible displacement in porous medium’, SIAM J. Math. Anal. 23 (1992), 1375–1392.CrossRefGoogle Scholar
[18]Feng, X., ‘Strong solutions to a nonlinear parabolic system modeling compressible miscible displacement in porous media’, Nonlinear Anal. 23 (1994), 1515–1531.CrossRefGoogle Scholar
[19]Feng, X., ‘On existence and uniqueness results for a coupled system modeling miscible displacement in porous media’, J. Math. Anal. Appl. 194 (1995), 883–910.CrossRefGoogle Scholar
[20]Frid, H., ‘Solution to the initial-boundary-value problem for the regularized Buckley-Leverett system’, Acta Appl. Math. 38 (1995), 239–265.CrossRefGoogle Scholar
[21]Frid, H. and Shelukin, V., ‘A quasi-linear parabolic system for three-phase capillary flow in porous media’, SIAM J. Math. Anal. 35 (2003), 1029–1041.CrossRefGoogle Scholar
[22]Gilbarg, D. and Trudinger, N.S., Elliptic partial differential equations of second order, Grundlehren der mathematischen Wissenschaften 224 (Springer-Verlag, Berlin, Heidelberg, New York, 1983).Google Scholar
[23]Hundsdorfer, W. and Verwer, J.G., Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Series in Computational Mathematics 33 (Springer-Verlag, Berlin, 2003).CrossRefGoogle Scholar
[24]Jerison, D. and Kenig, C.E., ‘The inhomogeneous Dirichlet problem in Lipschitz domains’, J. Funct. Anal. 130 (1995), 161–219.CrossRefGoogle Scholar
[25]Koval, E.J., ‘A method for predicting the performance of unstable miscible displacements in heterogeneous media’, SPEJ trans. AIME 228 (1963), 145–154.Google Scholar
[26]Kruzkov, S. N. and Sukorjanskii, S. M., ‘Boundary value problems for systems of equations of two-phase porous flow type; statement of the problems, questions of solvability, justification of approximate methods’, Math. USSR Sb. 33 (1977), 62–80.CrossRefGoogle Scholar
[27]Ladyzenskaja, O.A., Solonnikov, V.A. and Uralćeva, N.N., Linear and quasi-linear equations of parabolic type, Translation of Mathematical Monographs 23 (American Mathematical Society, Providence, RI, 1968).CrossRefGoogle Scholar
[28]Ladyzenskaja, O.A. and Uralćeva, N.N., Linear and quasi-linear elliptic equations (Academic Press, New York, London, 1968).CrossRefGoogle Scholar
[29]Marie, C.M., Multiphase flow in porous media (Culf Publishing Company, Houston, TX, 1981).Google Scholar
[30]McLaughlin, R.M. and Forest, M.G., ‘An anelastic, scale-separated model for mixing, with application to atmospheric transport phenomena’, Phys. Fluids 11 (1999), 880–892.CrossRefGoogle Scholar
[31]Meyers, N. G., ‘An L p estimate for the gradient of solutions of second order elliptic divergence equations’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1963), 189–206.Google Scholar
[32]Mikelić, A., ‘Regularity and uniqueness results for two-phase miscible flows in porous media’, Internat. Ser. Numer. Math. 114 (1993), 139–154.Google Scholar
[33]Pavliotis, G.A. and Kramer, P.R., ‘Homogenized transport by a spatiotemporal mean flow with small-scale periodic fluctuations’,in Proceedings of the fourth international conference on dynamical systems and differential equations (Discrete Contin. Dyn. Sys., AIMS, 2002), pp. 1–8.Google Scholar
[34]Peaceman, D.W., Fundamentals of numerical reservoir simulation (Elsevier, Amsterdam, 1977).Google Scholar
[35]Pearson, J.R.A. and Tardy, P.M.J., ‘Models for flows of non-newtonian and complex fluids through porous media’, J. Non-Newtonian Fluid Mech. 102 (2002), 447–473.CrossRefGoogle Scholar
[36]Sammon, P.H., ‘Numerical approximation for a miscible displacement process in porous media’, SIAM J. Numer. Anal. 23 (1986), 507–542.CrossRefGoogle Scholar
[37]Scheidegger, A.E., The physics of flow through porous media (Univ. Toronto Press, Toronto, Canada, 1974).Google Scholar
[38]Schroll, H.J. and Tveito, A., ‘Local existence and stability for a hyperbolic-elliptic system modeling two-phase reservoir flow’, Electron. J. Differential Equations 2000 (2000), 1–28.Google Scholar
[39]Simon, J., ‘Compact sets in the space L p(O, T;B)’, Ann. Math. Pura Appl. 146 (1987), 65–96.CrossRefGoogle Scholar
[40]Tallarek, U., Scheenen, T.W.J. and Van As, H., ‘Macroscopic heterogeneities in electroos-motic and pressure-driven flow through fixed beds at low column-to-particle diameter ratio’, J. Phys. Chem. B 105 (2001), 8591–8599.CrossRefGoogle Scholar
[41]Yannacopoulos, A.N. and Rowlands, G., ‘Effective drift velocities and effective diffusivities of swimming microorganisms in external flows’, J. Math. Biol. 39 (1999), 172–192.CrossRefGoogle ScholarPubMed