On a Solution of the Dirichlet–Cauchy Problem for the Barenblatt–Gilman Equation

N. Manakova   ; E. Bogatyreva

The Bulletin of Irkutsk State University. Series Mathematics, Tome 7 (2014), pp. 52-60

Résumé

We investigate the solvability of the Dirichlet–Cauchy problem for the Barenblatt–Gilman equation modeling the nonequilibrium countercurrent capillary impregnation. The feature of this model is the consideration of non-equilibrium effect — this becomes especially important when the process of impregnation takes a long time. Irregular and complex structure of the pore space does not allow to study the movement of liquids and gases therein by conventional methods of hydrodynamics. Hence the design and analysis of specific models describing these processes are required. The main equation of the model is nonlinear and not solvable for the derivative. This creates a significant difficulty in its consideration. The authors attribute the Barenblatt – Gilman equation to the wide class of Sobolev type equations. Sobolev type equations constitute an extensive area of nonclassical equations of mathematical physics. Research methods that are used in the work are initially emerged in the theory of semilinear Sobolev type equations. The equation is first considered in this context. The original problem is solved by the reduction in suitable functional spaces to the Cauchy problem for an abstract quasilinear Sobolev type equation with $s$-monotone and $p$-coercive operator. Existence theorems have been proven for generalized solutions of the abstract and the original problem.

Détail

Keywords: Barenblatt–Gilman equation, countercurrent capillary impregnation, quasilinear Sobolev type equation.

Bibliographie

[1] Barenblatt G. I., Gilman A. A., “Mathematical Model of the Countercurrent Capillary Impregnation”, Journal of Engineering Physics and Thermophysics, 52:3 (1987), 456–461 (in Russian)

[2] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev type equations and degenerate semigroups of operators, VSP, Utrecht–Boston–Köln–Tokio, 2003, 216 pp. | MR

[3] R. E. Showalter, “Nonlinear degenerate evolution equations and partial differential equations of mixed type”, SIAM J. Math. Anal., 6:1 (1975), 25–42 | DOI | MR

[4] A. B. Al'shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in nonlinear Sobolev-type equations, Walter de Gruyter, Berlin–N. Y., 2011, 648 pp. | MR

[5] Zagrebina S. A., “The Initial-Finite Problems for Nonclassical Models of Mathematical Physics”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 6:2 (2013), 5–24 (in Russian)

[6] Zamyshlyaeva A. A., “Stochastic Incomplete Linear Sobolev Type High-Ordered Equations with Additive White Noise”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 2012, no. 40 (299), issue 14, 73–82 (in Russian)

[7] Sagadeyeva M. A., “The Solvability of Nonstationary Problem of Filtering Theory”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 2012, no. 27 (286), issue 13, 86–98 (in Russian)

[8] Kozhanov A. I., “Linear Inverse Problems for a Class of Degenerate Equations of Sobolev”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 2012, no. 5 (264), issue 11, 33–42 (in Russian)

[9] Shestakov A. L., Sviridyuk G. A., “Optimal Measurement of Dynamically Distorted Signals”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 2011, no. 17 (234), issue 8, 70–75 (in Russian)

[10] Keller A. B., “The Algorithm for Solution of the Showalter-Sidorov Problem for Leontief Type Models”, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming Computer Software”, 2011, no. 4 (221), issue 7, 40–46 (in Russian)

[11] Sviridyuk G. A., Semenova I. N., “Solvability of the Inhomogenous Problem for the Generalized Boussinesq Filtration Equation”, Differential Equations, 24:9 (1988), 1607–1611 (in Russian) | MR

[12] Sviridyuk G. A., “One Problem for the Generalized Boussinesq Filtration Equation”, Russian Mathematics, 1989, no. 2, 55–61 (in Russian) | MR

[13] Manakova N. A., The Optimal Control Problems for Semilinear Sobolev Type Equations, SUSU publish. center, Chelyabinsk, 2012, 88 pp. (in Russian)

[14] Manakova N. A., Bogatyreva E. A., “Numerical Research of Processes in the Barenblatt-Gilman Model”, Bulletin of the Magnitogorsk State University. Mathematics, 2013, no. 15, 58–67 (in Russian)