Publication: Mathematical Models in Dynamics of Non-Newtonian Fluids and in Glaciology
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Oliveira, H.B de
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This paper deals with the study of some qualitative properties of solutions of mathematical models in non-Newtonian isothermal fluid flows and in theoretical glaciology. In the first type of models, we consider the extinction in a finite time of the solutions by using a global energy method. We prove that this property holds for pseudo-plastic fluids or for the general class of Newtonian and dilatant fluids, assumed the presence of a dissipation term (which may have an anisotropic nature and can vanish in, at most, one spatial direction). In the case of the ice sheet model in Glaciology (with a formulation involving a quasi-linear degenerate equation similar to the ones arising in non-Newtonian flows), we analyze the behavior of the free boundary (given by the support of the height h of the ice sheet) for different cases and according to the values of the ablation function and the initial hight. We use here some other energy methods of a local nature and so completely different to the method used in the first part of the paper.
CMNE CLAMCE 2007 Congresso Internacional em Métodos Numéricos em Engenharia, Porto, 13-15 junho 2007
S. N. Antontsev and H. B. de Oliveira, Navier-Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time. RIMS Kôkyûroku Bessatsu B1, Kyoto University (2007), pp. 21-42. S. N. Antontsev and H. B. de Oliveira. Finite time localized solutions of fluid problems with anisotropic dissipation. Internat. Ser. Numer. Math. 154, Birkhäuser (2006), pp. 23-32. S. N. Antontsev and H. B. de Oliveira. Localization of weak solutions for non-Newtonian fluid flows (Portuguese). Proceedings of the Congress Computational Methods in Engineering (CD-ROM). APMTAC and SEMNI, Laboratório Nacional S. N. Antontsev, J. I. Díaz and H. B. de Oliveira. On the confinement of a viscous fluid by means of a feedback external field. C. R. Mecanique 330 (2002), 797-802. S. N. Antontsev, J. I. Díaz and H. B. de Oliveira. Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem. J. Math. Fluid Mech., no. 4, 6 (2004),439-461. S. N. Antontsev, J. I. Díaz and H. B. de Oliveira. Stopping a viscous fluid by a feedback dissipative field. II. The stationary Navier-Stokes problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., no. 3-4, 15 (2004), 257-270. S. N. Antontsev, J. I. Díaz and H. B. de Oliveira. Stopping a viscous fluid by a feedback dissipative field: thermal effects without phase changing, Progr. Nonlinear Differential Equations Appl. 61, Birkhäuser (2005), 1-14. S. N. Antontsev, J. I. Díaz and S. I. Shmarev. Energy methods for free boundary problems, Progr. Nonlinear Differential Equations Appl. 48, Birkhäuser, 2002. S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov. Boundary value problems in mechanics of nonhomogeneous fluids. Studies in Mathematics and its Applications 22, NorthHolland, 1990. J.W. Barret and W.B. Liu. Finite element approximation of the parabolic p −laplacian. SIAM J. Numer. Anal. (2) 31 (1994), 413-428. Ph.Benilan, H. Brezis and M.G. Crandall. A semilinear equation in L1 (RN ), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523-555. F. Bernis. Compactness of the support for some nonlinear elliptic problems of arbitrary order in dimension n. Comm. Partial Differential Equations 9 (1984), no. 3, 271-312. F. Bernis. Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), no. 1-2, 1-19. M. Böhm. On a nonhomogeneous Bingham fluid. J. Differential Equations 60 (1985), 259–284. N. Calvo, J.I. Díaz, J. Durany, E. Schiavi and C. Vázquez. On a doubly nonlinear parabolic obstacle problem modelling ice sheet dynamics. SIAM J. Appl. Math. 63, 2 (2002), 683-707. J.I. Díaz. On the formation of the free boundary for the obstacle and Stefan problems via an energy method. CD-Rom Actas XVII CEDYA / VII CMA (L. Ferragut y A. Santos ed.), Servicio de Publicaciones de la Univ. de Salamanca, 2001. J.I. Díaz and M.A. Herrero. Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems. Proc. Roy. Soc. Edinburgh Sect. A 89 (1981), no. 3-4, 249-258. E. Fernández-Cara, F. Guillén and R. R. Ortega. Some theoretical results for viscoplastic and dilatant fluids with variable density. Nonlinear Anal., no. 6, 28 (1997), 1079–1100. E. Fernández-Cara, F. Guillén and R. R. Ortega. Some theoretical results concerning nonNewtonian fluids of the Oldroyd kind. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), no. 1, 26 (1998), 1–29. A.C. Fowler. Modelling ice sheet dynamics. Geophys. Astrophys. Fluid Dynam. 63, 1-4 (1992), 29-65. A.C. Fowler. Glaciers and ice sheets. The mathematics of models for climatology and environment. NATO ASI Ser. Ser. I Glob. Environ. Change 48 (1997), Springer, 301-336. K. Hutter. Theoretical Glaciology. D. Reidel Publishing Company, Dordrecht, 1982. O.A. Ladyzhenskaya. The mathematical theory of viscous incompressible flow. Mathematics and its Applications 2, Gordon and Breach, 1969. A.M. Meirmanov, V.V. Pukhnachov and S.I. Shmarev. Evolution equations and Lagrangian coordinates. Walter de Gruyter & Co., Berlin, 1997. C. Schoof. A variational approach to ice stream flow. J. Fluid Mech. 556 (2006), 227-251 . J. Simon. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal., no. 5, 21 (1990), 1093–1117. H. Sohr. The Navier-Stokes equations., Birkhäuser, Basel, 2001.