ABSTRACT In this paper we considered a weakly coupled system that consists of an elliptic equation and a degenerate parabolic equation, and it arises in the theory of flow of immiscible fluids in a porous medium ( see ([L] and [B1, Chapter 9; B2, Chapter 6; C, Chapter 6; L; S, Chapter 10]). The unknown functions u and v represent the pressure and the saturation respectively, subject to Darcy’s law and the Buckley-Leverett coupling. Due to the empirical nature of these laws no determination is possible on the structure of the degeneracy exhibited by the system. In this paper we established that the saturation is a locally continuous function in its space-time domain of definition, irrespective of the nature of the degeneracy of the principal part of the system. Questions of existence have been addressed in [ADB] for each of the indicated kind of boundary data, and to which we refer for the precise notion of weak solution. Recently this system has been investigated from a numerical and computational point of view (see [CE1, CE2, CEJS] and the references therein). However, a complete analytical treatment of the local behavior of its solutions to our knowledge is still lacking. By using the Kruzkov-Sukorjanski transformation [KS], the system can be separated into a parabolic equation for v, and an elliptic equation for u. The latter can be regarded as the equation of continuity of an idealized incompressible fluid formally replacing the mixture of the two fluids. The first regularity result can be found in [ADB], where it was proved that the saturations are con-tinuous functions provided one of the two permeabilities responsible for the degeneracy is mildly degenerate approaching the origin. These assumptions were relaxed in [DB], where a permeability was permitted to degenerate like a power. In either case no assumption was imposed on the behavior of the second degenerate point. If both permeabilities exhibit a power-like degeneracy at their degeneracy points, even with differ-ent powers, then in [U] the solutions are shown to be locally Hoelder continuous. The continuity of the saturation, with no degeneracy assumption was established in [DBV], in space dimension N = 2. In [DBV] the continuity was also established for $N\geq 3$ but in that work it was crucial that the principal part of the partial differential equation was exactly the Laplacian and that no lower order terms were present. In this paper we proved the continuity of the saturation in the indicated general physical setting that for any $N\geq 3$ was open. We proved that the presence of diffusion, however degenerate, suffices to insure the local continuity of the solutions over what might be expected by a transport process alone. The proof of this result is based on new technical results, that extend the techniques developed in [DBV] and improve a classical DeGiorgi Lemma [DG]. This approach was generalized in a recent paper [HV] in the context of singular quasilinear parabolic equations. From Google Scholar database it results that this paper was quoted 3 times. In this paper, one of the Authors, E. DiBenedetto is Professor in an American University (Vanderbilt University) REFERENCES [ADB] H.W. ALT and E. DIBENEDETTO, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 335– 392 [B1] J. BEAR, Dynamics of Fluids in Porous Media, Amer. Elsevier, New York, 1972. [B2] J. BEAR , Hydraulics of Groundwater, McGraw Hill, New York, 1979. [CE1] Z. CHEN and R.E. EWING, Degenerate two-phase incompressible flow. III. Sharp error estimates, J. Numer. Math. 90 (2001), no. 2, 215–240. [CE2] Z. CHEN and R.E. EWING , Degenerate two-phase incompressible flow. IV. Local refinement and domain decomposition, J. Sci. Comput. 18 (2003), no. 3, 329–360. [CEJS] Z. CHEN, R.E. EWING, Q. JIANG, and A.M. SPAGNUOLO, Degenerate two-phase incompressible flow. V. Characteristic finite element methods, J. Numer. Math. 10 (2002), no. 2, 87–107 [C] R.E. COLLINS, Flow of Fluids Through Porous Materials, Reinhold, New York, 1961 [DG] E. DE GIORGI Sulla differenziabilita’ e l’analiticita’ delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43. [DB] E. DIBENEDETTO, The flow of two immiscible fluids through a porous medium: regularity of the saturation, Theory and Applications of Liquid Crystals (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., vol. 5, Springer, New York, 1987, pp. 123–141. [DBV] E. DIBENEDETTO and V. VESPRI, On the singular equation $\beta(u)_t = \Delta u$ Arch. Rational Mech. Anal. 132 (1995), no. 3, 247–30 [KS] S. N. KRUZKOV and S. M. SUKORJANSKII, Boundary value problems for systems of equations of two-phase filtration type; formulation of problems, questions of solvability, justification of approximate methods, Mat. Sb. (N.S.) 104(146) (1977), no. 1, 69–88, 175–176 (Russian); English transl., Math. USSR-Sb. 33 (1977), no. 1, 62–80. [HV] E. HENRIQUES and V. VESPRI On the double degenerate equation $u_t-\mbox{div} a(x,t,u,\nablau)=b(x,t,u,\nablau)$. Nonlinear Analysis Series A: Theory, Methods & Applications. To appear [L] M.C. LEVERETT, Capillary behavior in porous solids, Trans. Amer. Inst. Mining and Metallurgicals Engrs. 142 (1941), 151–169. [S] A.E. SCHEIDEGGER, The Physics of Flow Through Porous Media, Third Edition, Univ. of Toronto Press, Toronto, Ontario, 1974. [U] J.M. URBANO, Hoelder continuity of local weak solutions for parabolic equations exhibiting two degeneracies, Adv. Differential Equations 6 (2001), no. 3, 327–358.

`http://hdl.handle.net/2158/595973`

Titolo: | Continuity of the saturation in the flow of two immiscible fluids in a porous medium |

Autori: | |

Autori: | E. DiBenedetto; U. Gianazza; V.Vespri |

Data di pubblicazione: | 2010 |

Rivista: | INDIANA UNIVERSITY MATHEMATICS JOURNAL |

Volume: | 59 |

Pagina iniziale: | 2029 |

Pagina finale: | 2064 |

Abstract: | ABSTRACT In this paper we considered a weakly coupled system that consists of an elliptic equation and a degenerate parabolic equation, and it arises in the theory of flow of immiscible fluids in a porous medium ( see ([L] and [B1, Chapter 9; B2, Chapter 6; C, Chapter 6; L; S, Chapter 10]). The unknown functions u and v represent the pressure and the saturation respectively, subject to Darcy’s law and the Buckley-Leverett coupling. Due to the empirical nature of these laws no determination is possible on the structure of the degeneracy exhibited by the system. In this paper we established that the saturation is a locally continuous function in its space-time domain of definition, irrespective of the nature of the degeneracy of the principal part of the system. Questions of existence have been addressed in [ADB] for each of the indicated kind of boundary data, and to which we refer for the precise notion of weak solution. Recently this system has been investigated from a numerical and computational point of view (see [CE1, CE2, CEJS] and the references therein). However, a complete analytical treatment of the local behavior of its solutions to our knowledge is still lacking. By using the Kruzkov-Sukorjanski transformation [KS], the system can be separated into a parabolic equation for v, and an elliptic equation for u. The latter can be regarded as the equation of continuity of an idealized incompressible fluid formally replacing the mixture of the two fluids. The first regularity result can be found in [ADB], where it was proved that the saturations are con-tinuous functions provided one of the two permeabilities responsible for the degeneracy is mildly degenerate approaching the origin. These assumptions were relaxed in [DB], where a permeability was permitted to degenerate like a power. In either case no assumption was imposed on the behavior of the second degenerate point. If both permeabilities exhibit a power-like degeneracy at their degeneracy points, even with differ-ent powers, then in [U] the solutions are shown to be locally Hoelder continuous. The continuity of the saturation, with no degeneracy assumption was established in [DBV], in space dimension N = 2. In [DBV] the continuity was also established for $N\geq 3$ but in that work it was crucial that the principal part of the partial differential equation was exactly the Laplacian and that no lower order terms were present. In this paper we proved the continuity of the saturation in the indicated general physical setting that for any $N\geq 3$ was open. We proved that the presence of diffusion, however degenerate, suffices to insure the local continuity of the solutions over what might be expected by a transport process alone. The proof of this result is based on new technical results, that extend the techniques developed in [DBV] and improve a classical DeGiorgi Lemma [DG]. This approach was generalized in a recent paper [HV] in the context of singular quasilinear parabolic equations. From Google Scholar database it results that this paper was quoted 3 times. In this paper, one of the Authors, E. DiBenedetto is Professor in an American University (Vanderbilt University) REFERENCES [ADB] H.W. ALT and E. DIBENEDETTO, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 3, 335– 392 [B1] J. BEAR, Dynamics of Fluids in Porous Media, Amer. Elsevier, New York, 1972. [B2] J. BEAR , Hydraulics of Groundwater, McGraw Hill, New York, 1979. [CE1] Z. CHEN and R.E. EWING, Degenerate two-phase incompressible flow. III. Sharp error estimates, J. Numer. Math. 90 (2001), no. 2, 215–240. [CE2] Z. CHEN and R.E. EWING , Degenerate two-phase incompressible flow. IV. Local refinement and domain decomposition, J. Sci. Comput. 18 (2003), no. 3, 329–360. [CEJS] Z. CHEN, R.E. EWING, Q. JIANG, and A.M. SPAGNUOLO, Degenerate two-phase incompressible flow. V. Characteristic finite element methods, J. Numer. Math. 10 (2002), no. 2, 87–107 [C] R.E. COLLINS, Flow of Fluids Through Porous Materials, Reinhold, New York, 1961 [DG] E. DE GIORGI Sulla differenziabilita’ e l’analiticita’ delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25–43. [DB] E. DIBENEDETTO, The flow of two immiscible fluids through a porous medium: regularity of the saturation, Theory and Applications of Liquid Crystals (Minneapolis, Minn., 1985), IMA Vol. Math. Appl., vol. 5, Springer, New York, 1987, pp. 123–141. [DBV] E. DIBENEDETTO and V. VESPRI, On the singular equation $\beta(u)_t = \Delta u$ Arch. Rational Mech. Anal. 132 (1995), no. 3, 247–30 [KS] S. N. KRUZKOV and S. M. SUKORJANSKII, Boundary value problems for systems of equations of two-phase filtration type; formulation of problems, questions of solvability, justification of approximate methods, Mat. Sb. (N.S.) 104(146) (1977), no. 1, 69–88, 175–176 (Russian); English transl., Math. USSR-Sb. 33 (1977), no. 1, 62–80. [HV] E. HENRIQUES and V. VESPRI On the double degenerate equation $u_t-\mbox{div} a(x,t,u,\nablau)=b(x,t,u,\nablau)$. Nonlinear Analysis Series A: Theory, Methods & Applications. To appear [L] M.C. LEVERETT, Capillary behavior in porous solids, Trans. Amer. Inst. Mining and Metallurgicals Engrs. 142 (1941), 151–169. [S] A.E. SCHEIDEGGER, The Physics of Flow Through Porous Media, Third Edition, Univ. of Toronto Press, Toronto, Ontario, 1974. [U] J.M. URBANO, Hoelder continuity of local weak solutions for parabolic equations exhibiting two degeneracies, Adv. Differential Equations 6 (2001), no. 3, 327–358. |

Handle: | http://hdl.handle.net/2158/595973 |

Appare nelle tipologie: | 1a - Articolo su rivista |

###### File in questo prodotto:

File | Descrizione | Tipologia | Licenza | |
---|---|---|---|---|

DGV_IUMJ_10-1.pdf | Versione Finale Referata | DRM non definito | Administrator |