Nonlinear Einstein paradigm of Brownian motion and localization property of solutions. (English) Zbl 1531.35249
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76R50 | Diffusion |
| 76A05 | Non-Newtonian fluids |
| 35K65 | Degenerate parabolic equations |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 35C06 | Self-similar solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B50 | Maximum principles in context of PDEs |
| 60J65 | Brownian motion |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
Keywords:
degenerate parabolic equations; diffusion processes; gradient-dependent diffusivity; localization property of solutionsReferences:
| [1] | E.DiBenedetto, Degenerate parabolic equations, Universitext, Springer‐Verlag, New York, NY, 1993. · Zbl 0794.35090 |
| [2] | J. L.Vázquez, The porous medium equation: Mathematical theory, Oxford University Press, Oxford, UK, 2007. · Zbl 1107.35003 |
| [3] | B.Straughan, Heat waves, Applied Mathematical Sciences, Vol. 177, Springer, New York, 2011. · Zbl 1250.80001 |
| [4] | J. B.Keller, Diffusion at finite speed and random walks, Proc. Natl Acad. Sci. USA101 (2004), 1120-1122. · Zbl 1069.60066 |
| [5] | B. H.Gilding and R.Kersner, Travelling waves in nonlinear diffusion‐convection reaction, Progress in Nonlinear Differential Equations and Their Applications, Vol. 60, Birkhäuser, Basel, 2004. · Zbl 1073.35002 |
| [6] | G. I.Barenblatt, On some unsteady fluid and gas motions in a porous medium, Prik. Mat. Mekh. (PMM)16 (1952), 67-78. In Russian. · Zbl 0049.41902 |
| [7] | J. R.Philip, Flow in porous media, Annu. Rev. Fluid Mech.2 (1970), 177-204. |
| [8] | A.Oron, S. H.Davis, and S. G.Bankoff, Long‐scale evolution of thin liquid films, Rev. Mod. Phys.69 (1997), 931-980. |
| [9] | A. L.Bertozzi and M.Pugh, The lubrication approximation for thin viscous films: Regularity and long‐time behavior of weak solutions, Commun. Pure Appl. Math.49 (1996), 85-123. · Zbl 0863.76017 |
| [10] | T. P.Witelski, Nonlinear dynamics of dewetting thin films, AIMS Mathematics5 (2020), 4229-4259. · Zbl 1484.76015 |
| [11] | Ya. B.Zel’dovich and Yu. P.Raizer, Physics of shock waves and high‐temperature hydrodynamic phenomena, Vol. II, Academic Press, New York, 1967. |
| [12] | E.Celik, L.Hoang, A.Ibragimov, and T.Kieu, Fluid flows of mixed regimes in porous media, J. Math. Phys.58 (2017), 23102. · Zbl 1357.76086 |
| [13] | V.Ciriello, S.Longo, L.Chiapponi, and DiFederico, Porous gravity currents: A survey to determine the joint influence of fluid rheology and variations of medium properties, Adv. Water Res.92 (2016), 105-115. |
| [14] | V.Di Federico, S.Longo, S. E.King, L.Chiapponi, D.Petrolo, and V.Ciriello, Gravity‐driven flow of Herschel-Bulkley fluid in a fracture and in a 2D porous medium, J. Fluid Mech.821 (2017), 59-84. · Zbl 1383.76442 |
| [15] | A. A.Ghodgaonkar and I. C.Christov, Solving nonlinear parabolic equations by a strongly implicit finite difference scheme: Applications to the finite speed spreading of non‐newtonian viscous gravity currents, Applied wave mathematics II, A.Berezovski (ed.) and T.Soomere (ed.), (eds.), Mathematics of Planet Earth, Vol. 6, Springer Nature, Cham, Switzerland, 2019, pp. 305-342. · Zbl 1443.76162 |
| [16] | A. S.Kalashnikov, Some problems of the qualitative theory of non‐linear degenerate second‐order parabolic equations, Russ. Math. Surv.42 (1987), 169-222. · Zbl 0642.35047 |
| [17] | A.Einstein, Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen, Ann. Phys. (Leipzig)322 (1905), 549-560. · JFM 36.0975.01 |
| [18] | A.Einstein, Investigations on the theory of the Brownian movement, Dover Publications, Mineola, NY, 1956. Edited by R. Fürth, Translated by A. D. Cowper. · Zbl 0071.41205 |
| [19] | R. E.Cunningham and R. J. J.Williams, Diffusion in gases and porous media, Springer, Boston, MA, 1980. |
| [20] | C. W.Gardiner, Stochastic methods: A handbook for the natural and social sciences, 4th ed., Springer Series in Synergetics, Vol. 13, Springer‐Verlag, Berlin, 2009. · Zbl 1181.60001 |
| [21] | I. C.Christov, A.Ibraguimov, and R.Islam, Long‐time asymptotics of non‐degenerate non‐linear diffusion equations, J. Math. Phys.61 (2020), 81505. · Zbl 1454.35030 |
| [22] | V.Méndez, D.Campos, and F.Bartumeus, Stochastic foundations in movement ecology: Anomalous diffusion, front propagation and random searches, Springer Series in Synergetics, Springer‐Verlag, Berlin/Heidelberg, 2014. · Zbl 1295.92010 |
| [23] | R. G.Barle and D. R.Sherbert, Introduction to real analysis, 4th ed., John Wiley & Sons, New York, NY, 2010. |
| [24] | W. W.Krückels, On gradient dependent diffusivity, Chem. Eng. Sci.28 (1973), 1565-1576. |
| [25] | A. M.Ilyin, A. S.Kalashnikov, and O. A.Oleynik, Linear second‐order partial differential equations of the parabolic type, J. Math. Sci.108 (2002), 435-542. · Zbl 0988.35001 |
| [26] | E. M.Landis, Second order equations of elliptic and parabolic type, Translations of Mathematical Monographs, Vol. 171, American Mathematical Society, Providence, RI, 1998. · Zbl 0895.35001 |
| [27] | M. G.Crandall, H.Ishii, and P.‐L.Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc.27 (1992), 1-67. · Zbl 0755.35015 |
| [28] | S.Kamin and J. L.Vazquez, Fundamental solutions and asymptotic behaviour for the [( p \]\)‐Laplacian equation, Rev. Mat. Iberoam.4 (1988), 339-354. · Zbl 0699.35158 |
| [29] | A.Tedeev and V.Vespri, Optimal behavior of the support of the solutions to a class of degenerate parabolic systems, Interfaces Free Bound.17 (2015), 143-156. · Zbl 1331.35198 |
| [30] | E.DiBenedetto, U.Gianazza, and V.Vespri, Harnack’s inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics, Springer, New York, NY, 2012. |
| [31] | O. A.Ladyženskaja, V. A.Solonnikov, and N. N.Ural’ceva, Linear and quasi‐linear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, RI, 1968. · Zbl 0174.15403 |
| [32] | D. G.Aronson, L. A.Caffarelli, and S.Kamin, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal.14 (1983), 639-658. · Zbl 0542.76119 |
| [33] | G. I.Barenblatt and Y. B.Zel’dovich, Self‐similar solutions as intermediate asymptotics, Annu. Rev. Fluid Mech.4 (1972), 285-312. · Zbl 0249.76001 |
| [34] | G. I.Barenblatt, Scaling, self‐similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, Vol. 14, Cambridge University Press, New York, 1996. · Zbl 0907.76002 |
| [35] | J.Eggers and M. A.Fontelos, Singularities: Formation, structure and propagation, Cambridge Texts in Applied Mathematics, Vol. 53, Cambridge University Press, New York, NY, 2015. · Zbl 1335.76002 |
| [36] | R. E.Pattle, Diffusion from an instantaneous point source with a concentration‐dependent coefficient, Quart. J. Mech. Appl. Math.12 (1959), 407-409. · Zbl 0119.30505 |
| [37] | G. I.Barenblatt, On self‐similar motions of compressible fluid in a porous medium, Prik. Mat. Mekh. (PMM)16 (1952), 679-698. In Russian. · Zbl 0047.19204 |
| [38] | B. H.Gilding and L. A.Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl.55 (1976), 351-364. DOI 10.1016/0022‐247X(76)90166‐9 · Zbl 0356.35049 |
| [39] | S.Kamin, L. A.Peletier, and J. L.Vazquez, Classification of singular solutions of a nonlinear heat equation, Duke Math. J.58 (1989), 601-615. · Zbl 0701.35083 |
| [40] | E. M.Landis, On the “dead zone” for semilinear degenerate elliptic inequalities, Differ. Uravn.29 (1993), no. 3, 414-423. http://mi.mathnet.ru/de8054. [Translated as Differ. Equ., 29 (1993), pp. 355-363]. · Zbl 0792.35014 |
| [41] | E. M.Landis, A new proof of E. De Georgi’s theorem, Tr. Mosk. Mat. Obs.16 (1967), 319-328. http://mi.mathnet.ru/mmo188. In Russian. · Zbl 0182.43403 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
