Journal of the Korean Society for Industrial and Applied Mathematics

  • 제26권2호
  • /
  • Pages.108-120
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  • 2022
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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FINITE SPEED OF PROPAGATION IN DEGENERATE EINSTEIN BROWNIAN MOTION MODEL

  • 투고 : 2022.01.17
  • 심사 : 2022.05.16
  • 발행 : 2022.06.25
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초록

We considered qualitative behaviour of the generalization of Einstein's model of Brownian motion when the key parameter of the time interval of free jump degenerates. Fluids will be characterised by number of particles per unit volume (density of fluid) at point of observation. Degeneration of the phenomenon manifests in two scenarios: a) flow of the fluid, which is highly dispersing like a non-dense gas and b) flow of fluid far away from the source of flow, when the velocity of the flow is incomparably smaller than the gradient of the density. First, we will show that both types of flows can be modeled using the Einstein paradigm. We will investigate the question: What features will particle flow exhibit if the time interval of the free jump is inverse proportional to the density and its gradient ? We will show that in this scenario, the flow exhibits localization property, namely: if at some moment of time t0 in the region, the gradient of the density or density itself is equal to zero, then for some T during time interval [t0, t0 + T] there is no flow in the region. This directly links to Barenblatt's finite speed of propagation property for the degenerate equation. The method of the proof is very different from Barenblatt's method and based on the application of Ladyzhenskaya - De Giorgi iterative scheme and Vespri - Tedeev technique. From PDE point of view it assumed that solution exists in appropriate Sobolev type of space.

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참고문헌

  1. Editors of Encyclopedia, Mean free path-Encyclopedia Britannica, 2007. https://www.britannica.com/science/mean-freepath.
  2. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, 2010.
  3. G. I. Barenblatt, On some unsteady fluid and gas motions in a porous medium, PMM journal of Applied Mathematics and Mechanics, 16 (1952), 67-78.
  4. G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics, Cambridge Texts in Applied Mathematics 14, Cambridge University Press, Cambridge, 1996.
  5. G. I. Barenblatt, On self-similar motions of compressible fluid in a porous medium, PMM journal of Applied Mathematics and Mechanics, 16 (1952),679-698.
  6. R. Farmani, R. Azina and R. Fateh, Analysis of Pre-Darcy flow for different liquids and gases, Journal of Petroleum Science and Engineering, 168 (2018), 17-31. https://doi.org/10.1016/j.petrol.2018.05.004
  7. L. Bloshanskaya, A. Ibragimov, F. Siddiqui and M. Y. Soliman, Productivity Index for Darcy and pre-/post-Darcy Flow (Analytical Approach), Journal of Porous Media, 20 (2017), 769-786. https://doi.org/10.1615/jpormedia.v20.i9.10
  8. R.P. Chhabra and J.F. Richardson, Non-newtonian flow in the process industries: Fundamentals and engineering applications, Reed Educational and Professional Publishing Ltd, Oxford, UK, 1999.
  9. A. Einstein, Investigations on the Theory of the Brownian Movement, edited by R. Furth, Dover Publications, New York, translated by A.D. Cowper, 1956.
  10. E. M. Landis, On the dead zone for semilinear degenerate elliptic inequalities, Differ Uravn, 29 (1993), 414-423.
  11. B. Brogliato and A. Tanwani, Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability, Society for Industrial and Applied Mathematics, 62(1) (2020), 3-129.
  12. A.F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications 18, Springer, Netherlands, 1988.
  13. G.I. Barenblatt, V.M. Entov and V.M. Ryzhik, Theories of Fluid Flows Through Natural Rocks, Theory and Applications of Transport in Porous Media 3, Dordrecht ; Boston : Kluwer Academic Publishers, 1990.
  14. Y. B. Zeldovich and G. I. Barenblatt, On the asymptotic properties of self-similar solutions of the equations of unsteady gas filtration, Doklady, USSR Academy of Sciences, 118 (1958), 671-674.
  15. J. L. Padgett, Y. Geldiyev, S. Gautam, W. Peng, Y. Mechref and A. Ibraguimov, Object classification in analytical chemistry via data-driven discovery of partial differential equations, Computational and Mathematical Methods, 3 (2021), e1164.
  16. R. Ewing, A. Ibragimov and R. Lazarov, Domain decomposition algorithm and analytical simulation of coupled flow in reservoir/well system, KSIAM, 5(2) (2001), 71-99.
  17. I. C. Christov, A. Ibraguimov, R. Islam, Long-time asymptotics of non-degenerate non-linear diffusion equations, Journal of Mathematical Physics, 61 (2020), 081505. https://doi.org/10.1063/5.0005339
  18. M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bulletin of the American Mathematical Society, 27 (1992), 1-67. https://doi.org/10.1090/S0273-0979-1992-00266-5
  19. A. Tedeev and V. Vespri, Optimal behavior of the support of the solutions to a class of degenerate parabolic systems, Interfaces and Free Boundaries, 17 (2015), 143-156. https://doi.org/10.4171/IFB/337
  20. E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics 165 Springer, New York, 2012.
  21. O.A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs 23, American Mathematical Society, Providence,RI, 1968.
  22. S. Kamin and J. L. Va'squez, Fundamental Solutions and Asymptotic Behaviour for the p-Laplacian Equation, Revista Matematica Iberoamericana, 4 (1988), 339-354.
  23. E. de Giorgi, Sulla differenziabilita e l'analiticit ' a delle estremali degli integrali multipli regolari, Matematica, 4 (1960), 23-38.
  24. E. M. Landis, Second Order Equations of Elliptic and Parabolic Type, Translations of Mathematical monographs 171, American Mathematical Society, Providence, RI, 1997.