On a free boundary problem for a strongly degenerate quasi-linear parabolic equation with an application to a model of pressure filtration

SIAM Journal on Mathematical Analysis

Volumn 34, Issue 3, 2003, Pages 611-635

  Bürger, Raimund ^a   Frid, Hermano ^b   Karlsen, Kenneth H ^c  

^a UNIVERSITY OF STUTTGART   (Germany)

^b IMPA   (Brazil)

^c UNIVERSITY OF BERGEN   (Norway)

Author keywords

Divergence measure field; Free boundary problem; Pressure filtration; Strongly degenerate parabolic equation

Indexed keywords

EID: 13244256081     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141002401007     Document Type: Article

References (28)

1. G. ANZELLOTTI, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4), 135 (1983), pp. 293-318.
2. C. BARDOS, A.Y. LE ROUX, AND J.C. NEDELEC, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), pp. 1017-1034.
3. R. BÜRGER, F. CONCHA, AND K.H. KARLSEN, Phenomenological model of filtration processes: 1. Cake formation and expression, Chem. Engrg. Sci., 56 (2001), pp. 4537-4553.
4. R. BÜRGER, S. EVJE, AND K.H. KARLSEN, On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes, J. Math. Anal. Appl., 247 (2000), pp. 517-556.
5. R. BÜRGER AND W.L. WENDLAND, Existence, uniqueness, and stability of generalized solutions of an initial-boundary value problem for a degenerating quasilinear parabolic equation, J. Math. Anal. Appl., 218 (1998), pp. 207-239.
6. R. BÜRGER AND W.L. WENDLAND, Entropy boundary and jump conditions in the theory of sedimentation with compression, Math. Methods Appl. Sci., 21 (1998), pp. 865-882.
7. R. BÜRGER, W.L. WENDLAND, AND F. CONCHA, Model equations for gravitational sedimentation-consolidation processes, ZAMM Z. Angew. Math. Mech., 80 (2000), pp. 79-92.
8. M.C. BUSTOS, F. CONCHA, R. BÜRGER, AND E.M. TORY, Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory, Kluwer Academic, Dordrecht, The Netherlands, 1999.
9. G.-Q. CHEN AND H. FRID, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), pp. 89-118.
10. G.-Q. CHEN AND H. FRID, The theory of divergence-measure fields and applications, Bol. Soc. Brasil. Mat. (N.S.), 32 (2001), pp. 401-433.
11. L.C. EVANS AND R.F. GARIEPY, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.
12. P. GARRIDO, F. CONCHA, AND R. BÜRGER, Application of the unified model of solid-liquid separation of flocculated suspensions to experimental results, in Proceedings of the Sixth Southern Hemisphere Meeting on Mineral Technology, Rio de Janeiro, Brazil, 2001, Vol. 1, A.B. da Luz, P.S.M. Scares, M.L. Torem, and R.B.E. Trinidads, eds., CETEM/MCT, Rio de Janeiro, Brazil, 2001, pp. 117-122.
13. K.H. KARLSEN AND N.H. RISEBRO, Corrected operator splitting for nonlinear parabolic equations, SIAM J. Numer. Anal., 37 (2000), pp. 980-1003.
14. S.N. KRUŽKOV, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), pp. 217-243.
15. O.A. LADYŽENSKAJA, V.A. SOLONNIKOV, AND N.N. URAL'CEVA, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, AMS, Providence, RI, 1968.
16. K.A. LANDMAN AND W.B. RUSSEL, Filtration at large pressures for strongly flocculated suspensions, Phys. Fluids A, 5 (1993), pp. 550-560.
17. J. MÁLEK, J. NEČAS, M. ROKYTA, AND M. RUŽIČKA, Weak and Measure-Valued Solutions to Evolutionary PDEs, Chapman &: Hall, London, 1996.
18. C. MASCIA, A. PORRETTA, AND A. TERRACINA, Non-homogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal., 163 (2002), pp. 87-124.
19. F. OTTO, First Order Equations with Boundary Conditions, Preprint, SFB 256, University of Bonn, Germany, 1992.
20. F. OTTO, Ein Randwertproblem für Erhaltungssätze, Doctoral thesis, University of Bonn, Germany, 1993.
21. F. OTTO, Initial-boundary value problem for a scalar conservation law, C.R. Acad. Sci. Paris Sér. I Math., 322 (1996), pp. 729-734.
22. S. SAKS, Theory of the Integral, Monografie Matematyczne, Drukarna Uniw. Jagiellonskiego, Warsaw, Poland, G.E. Stechert, New York, 1937.
23. J. WLOKA, Partielle Differentialgleichungen, Teubner Verlag, Stuttgart, 1982.
24. Z. WU, A note on the first boundary value problem for quasilinear degenerate parabolic equations, Acta Math. Sci. (English Ed.), 4 (1982), pp. 361-373.
25. Z. WU, A Boundary Value Problem for Quasilinear Degenerate Parabolic Equations, MRC Technical Summary Report #2484, University of Wisconsin, Madison, WI, 1983.
26. Z. WU AND J. WANG, Some results on quasilinear degenerate parabolic equations of second order, in Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 3, Science Press, Beijing, Gordon and Breach, New York, 1982, pp. 1593-1609.
27. Z. WU, J. ZHAO, J. YIN, AND H. LI, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001.
28. J. ZHAO AND Y. LI, A free boundary problem for quasilinear degenerate parabolic equations with general degeneracy, Acta Math. Sinica (N.S.), 6 (1990), pp. 364-382.