Volume 7, no. 3Pages 60 - 68

On the Well-Posedness of Some Problems of Filtration in Porous Media

M.N. Nebolsina, Al Khazraji S.H.M.
Using the theory of semigroups of linear transformations, we establish the uniform well-posedness of initial-boundary value problems for a class of integrodifferential equations in bounded and half-bounded regions describing the processes of nonstationary filtration of squeezing liquid in porous media.
Babenko considered a particular case of these equations on the semi-infinite straight line with Dirichlet condition on the boundary. In that work it was required to find the pressure gradient on the boundary, and the answer is obtained by the formal application of fractional integro-differentiation while ignoring the question of continuous dependence on the intial data. The solution is expressed as a formal series involving an unbounded operator, whose convergence is not discussed.
The theory of strongly continuous semigroups of transformations enables us to establish the uniform well-posedness of the Dirichlet and Neumann problems for both finite and infinite regions. It enables us to calculate the pressure gradient on the boundary in the case of the Dirichlet problem and the boundary value of the solution in the case of the Neumann problem. We also prove that the solution is stable with respect to the initial data.
Full text
Keywords
filtration processes; porous media; well-posed problem; C_0-semigroups; fractional powers of operators.
References
1. Babenko Yu.I. Teplomassoobmen, metody rascheta teplovyh i diffuzionnyh potokov [ Heat and Mass Transfer. The Method of Calculation of Heat and Diffusion Currents]. Leningrad, Chemistry, 1986.
2. Knyazyuk A.V. Granichnye znacheniya evolyutsionnykh uravneniy v banakhovom prostranstve [Boundary Values of Evolution Equations in a Banach Space. The Dissertation for Scientific Degree of the Candidate of Physical Mathematical Sciences]. Kiev, 1985. 115 p.
3. Krejn S.G. Lineynye differential'nye uravneniya v banakhovom prostranstve [Linear Differential Equations in Banach Spaces]. Moscow, Nauka, 1967. 464 p.
4. Kostin D.V. [On the Third Boundary-Value Problem for Equations of Elliptic Type in a Banach Space on R^{+}]. Materials of Voronezh Spring Mathematical School 'Pontryagin Readings-XXIII', Voronezh, VSU, 2012, pp. 97.
5. Lavrent'ev M.A, Shabat B.V. Metody teorii funktsiy kompleksnogo peremennogo [Methods of the Theory of Functions of a Complex Variable]. Moscow, Nauka, 1973.
6. Mumford D. Tata Lectures on Theta. Boston, Basel, Berlin, Birkhlfuser, 1983. DOI: 10.1007/978-1-4899-2843-6
7. Kostin V.A., Nebol'sina M.N. Well-Posedness of Boundary Value Problems for a Second-Order Equation. Doklady Mathematics, 2009, vol. 80, no. 2, pp. 650-652. DOI: 10.1134/S1064562409050044
8. Nebolsina M.N. Issledovanie korrektnoy razreshimosti nekotoryh matematicheskih modeley teplomassoperenosa metodom S.G.Kreina [The Research of Correct Solvability of Some Mathematical Models of Heat and Mass Transfer Method S.G. Krein. The Dissertation for Scientific Degree of the Candidate Physical and Mathematical Sciences]. Voronezh, VSU, 2009. 102 p.