Study of some mathematical models for nonstationary filtration processes


  • N.L. Gol’dman


parabolic equations
boundary value problems
Holder spaces
Rothe method
filtration processes


We consider some mathematical models connected with the study of nonstationary filtration processes in underground hydrodynamics. These models involve nonlinear problems for parabolic equations with unknown source functions. One of the problems is a system consisting of a boundary value problem of the first kind and an equation describing a time dependence of the sought source function. In the other problem, the corresponding system is distinguished from the first one by boundary conditions of the second kind. These problems essentially differ from usual boundary value problems for parabolic equations. The aim of our study is to establish conditions of unique solvability in a class of smooth functions for the considered nonlinear parabolic problems. The proposed approach involves the proof of a priori estimates for the Rothe method.





Section 1. Numerical methods and applications

Author Biography

N.L. Gol’dman

Lomonosov Moscow State University,
Research Computing Center
Leninskie Gory, Moscow, 119991, Russia
• Leading Researcher


  1. G. I. Barenblatt, Yu. P. Zheltov, and I. N. Kochina, “Basic Concepts in the Theory of Seepage of Homogeneous Liquids in Fissured Rocks,” Prikl. Mat. Mekh. 24 (5), 852-864 (1960) [J. Appl. Math. Mech. 24 (5), 1286-1303 (1960)].
  2. G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Theory of Fluid Flows through Natural Rocks (Nedra, Moscow, 1984; Kluwer, Dordrecht, 1990).
  3. D. A. Gubaidullin and R. V. Sadovnikov, “Application of Parallel Algorithms for Solving the Problem of Fluid Flow to Wells with Complicated Configurations in Fractured Porous Reservoirs,” Vychisl. Metody Programm. 8, 244-251 (2007).
  4. G. G. Chernyi, Selected Works (Nauka, Moscow, 2009) [in Russian].
  5. M. H. Khairullin, A. I. Abdullin, P. E. Morozov, and M. N. Shamsiev, “The Numerical Solution of the Inverse Problem for the Deformable Porous Fractured Reservoir,” Mat. Model. 20 (11), 35-40 (2008).
  6. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; AMS Press, Providence, 1968).
  7. A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, Englewood Cliffs, 1964; Mir, Moscow, 1968).
  8. N. L. Gol’dman, Inverse Stefan Problems (Kluwer, Dordrecht, 1997).
  9. N. L. Gol’dman, Inverse Stefan Problems. Theory and Methods of Solution (Mosk. Gos. Univ., Moscow, 1999) [in Russian].
  10. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977; Pergamon, New York, 1982).
  11. S. N. Kruzhkov, “A Priori Estimate for the Derivative of a Solution to a Parabolic Equation,” Vestn. Mosk. Univ., Ser. 1: Mat. Mekh., No. 2, 41-48 (1967).
  12. N. L. Gol’dman, “Boundary Value Problems for a Quasilinear Parabolic Equation with an Unknown Coefficient,” J. Differ. Equations 266 (8), 4925-4952 (2019).