On the solvability of a variational inequality in the filtration theory

M. F. Pavlova ; E. V. Rung

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 161 (2019) no. 4, pp. 552-568

Résumé

In this paper, we proved the generalized solvability of a problem describing the process of unsteady saturated-unsaturated fluid filtration in a porous medium with the condition of unilateral permeability to parts of the boundary. It should be noted that the variational inequality that arises in this case is a variational inequality of a variable type: in the saturated filtration zone – elliptical and parabolic – otherwise. In the generalized formulation of the problem under consideration, a classical transition based on the Kirchhoff transform to an equivalent variational problem that is more convenient for research was used. In this paper, we considered the most interesting case, from the point of applications, when the Kirchhoff transform maps the real axis into a semi-axis bounded below: $[-\gamma,+\infty).$ It is assumed that the value of the Kirchhoff transform at a point $-\gamma$ is zero. Along with the original problem with restriction, we considered the so-called “predefined problem” without restrictions $u(x,t) \geq -\gamma$, the solution of which on the set $(-\infty,-\gamma)$ is assumed to be zero. Definitions of generalized solutions to these problems in the form of variational inequalities were given. The proof of the existence theorem for a generalized solution of the “predefined problem” was carried out using the methods of half-sampling and penalty. In conclusion, it was proved that the solution to the “predetermined problem” is the solution to the original one.

Détail

Mots-clés : filtration
Keywords: variational inequality, Kirchhoff transform, penalty half-sampling method, Galerkin method.

Bibliographie

[1] Pavlova M.F, Rung E. V., “On the existence of a generalized solution of the saturated-unsaturated filtration problem”, Differ. Equations, 54:3 (2018), 352–362 | DOI | MR | Zbl

[2] Collins R., Flow of Fluids through Porous Materials, Van Nostrand, New York, 1961, 352 pp.

[3] Alt H. W., Luckhaus S., “Quasilinear elliptic-parabolic differential equations”, Math. Z., 183:8 (1983), 311–341 | DOI | MR | Zbl

[4] Morel-Seytoux H. J., Meyer P. D., Nachabe M., Touma J., van Genuchten M. T., Lenhard R. J., “Parameter equivalence for the Brooks–Corey and van Genuchten soil characteristics: Preserving the effective capillary drive”, Water Resour. Res., 32:58 (1996), 1251–1258 | DOI

[5] Pop I. S., Yong W.-A., “A numerical approach to degenerate parabolic equations”, Numer. Math., 92 (2002), 357–381 | DOI | MR | Zbl

[6] Diersch H.-J.G., Perrochet P., “On the primary variable switching technique for simulating unsaturated-saturated flows”, Adv. Water Resour., 23:3 (1999), 271–301 | DOI

[7] Radu F., Pop I. S., Knabner P., “Order of convergence estimates in time and space for an implicit Euler, mixed finite element discretization of Richard's equation by equivalence of mixed and conformal approach”, Proc. Conf. Algoritmy 2002, eds. A. Handlovicova, Z. Kriva, K. Mikula, D. Sevcovic, 2002, 58–65

[8] Akhtareev A. A., Dautov R. Z., “Mixed variable method for simulating saturated-unsaturated flows”, Uchenye Zapiski Kazanskogo Gosudarstvennogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 149, no. 4, 2007, 58–72 (In Russian)

[9] Williams G. A., Miller C. T., Kelley C. T., “Transformation approaches for simulating flow in variably saturated porous media”, Water Res. Resear., 36:4 (2000), 923–934 | DOI

[10] Berninger H., Domain decomposition methods for elliptic problems with jumping nonlinearities and application to the Richards equation, Dissertation, Freie Univ., Berlin, 2008, 272 pp. | DOI

[11] Otto F., “$L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations”, J. Differ. Equations, 131:1 (1996), 20–38 | DOI | MR | Zbl

[12] Otto F., “$L^1$-contraction and uniqueness for unstationary saturated-unsaturated porous media flow”, Adv. Math. Sci. Appl., 7:2 (1997), 537–553 | MR | Zbl

[13] Van Duyn C. J., Peletier L. A., “Nonstationary filtration in partially saturated porous media”, Arch. Ration. Mech. Anal., 78:2 (1982), 173–198 | DOI | MR | Zbl

[14] Alt H. W., Luckhaus S., Visintin A., “On nonstationary flow through porous media”, Ann. Mat. Pura ed Appl., 136 (1984), 303–316 | DOI | MR | Zbl

[15] Pavlova M. F., Shemuranova Ye.V. Existence of a weak solution for a problem of unsaturated filtration consolidation, Russ. Math., 45:10 (2001), 54–64 | MR | Zbl

[16] Pavlova M. F., “Investigation of the well-posedness of a filtration consolidation problem with incomplete saturation”, Differ. Equations, 34:7 (1998), 967–976 | MR | Zbl

[17] Lions J.-L., Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod, Paris, 1969, 554 pp. (In French) | MR

[18] Karchevskii M. M., Pavlova M. F., Equations of Mathematical Physics. Additional Chapters, Lan', St. Petersburg, 2016, 276 pp. (In Russian)

[19] Gilbarg D., Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983, 464 pp. | MR | Zbl

[20] Gajewski H., Gröger K., Zacharias K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974, 281 pp. (In German) | MR | Zbl