×

Pattern formation in a flux limited reaction-diffusion equation of porous media type. (English) Zbl 1388.35094

In this very interesting work, the authors investigate pattern formation in porous media. More specifically, they examine the existence and properties of traveling wave solutions for a non-linear Fokker-Planck equation with flux limited diffusion, of the form \[ u_t=\nu \text{div}\Big(\frac{u^m\nabla u}{\sqrt{|u|^2+\frac{\nu^2}{c^2}|\nabla u|^2}}\Big)+F(u), \] posed in \((0,T)\times \mathbb{R}^n\), where \(m>1\), \(\nu\) is a kinematic viscosity, and \(c>0\) is a characteristic speed; the nonlinearity \(F(u)\) is a Lipschitz continuous function satisfying \[ F(0)=F(1)=0. \] In particular, one-dimensional traveling wave solutions \[ u(x-\sigma t), \] with constant speed \(\sigma>0\), and range in \([0, 1]\) are considered. Depending on the wave speed, existence of classical traveling waves is established, while for \[ \sigma_{\text{ent}}\leq\sigma<\sigma_{\text{smooth}}, \] discontinuous waves are developed that resemble to hyperbolic shock waves. In the case of \(\sigma = \sigma_{\text{ent}}\), the discontinuous solutions are supported on the half line and correspond to propagation of information at finite speed.

MSC:

35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35K67 Singular parabolic equations
34Cxx Qualitative theory for ordinary differential equations
70Kxx Nonlinear dynamics in mechanics
35B60 Continuation and prolongation of solutions to PDEs
37Dxx Dynamical systems with hyperbolic behavior
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
35Q99 Partial differential equations of mathematical physics and other areas of application
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (2000) · Zbl 0957.49001
[2] Andreu, F., Caselles, V., Mazón, J.M.: A Strongly Degenerate Quasilinear Elliptic Equation. Nonlinear Anal. TMA 61, 637-669 (2005) · Zbl 1190.35100 · doi:10.1016/j.na.2004.11.020
[3] Andreu, F., Caselles, V., Mazón, J.M.: A Fisher-Kolmogorov-Petrovskii-Piskunov equation with finite speed of propagation. J. Differ. Equ. 248, 2528-2561 (2010) · Zbl 1198.35121 · doi:10.1016/j.jde.2010.01.005
[4] Andreu, F., Caselles, V., Mazón, J.M.: The Cauchy Problem for a Strongly Degenerate Quasilinear Equation. J. Eur. Math. Soc. 7, 361-393 (2005) · Zbl 1082.35089 · doi:10.4171/JEMS/32
[5] Andreu, F., Caselles, V., Mazón, J.M., Soler, J., Verbeni, M.: Radially symmetric solutions of a tempered diffusion equation. A porous media flux-limited case. SIAM J. Math. Anal. 44, 1019-1049 (2012) · Zbl 1276.35109 · doi:10.1137/110840297
[6] Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusions arising in population genetics. Adv. Math. 30, 33-76 (1978) · Zbl 0407.92014 · doi:10.1016/0001-8708(78)90130-5
[7] Aronson, D.G.: Density dependent interaction diffusion systems. In: Proceedings of the Advanced Seminar on Dynamics and Modeling of Reactive Systems, Academic Press, New York (1980) · Zbl 1181.80001
[8] Anzellotti, G.: Pairings Between Measures and Bounded Functions and Compensated Compactness. Ann. di Matematica Pura et Appl. IV 135, 293-318 (1983) · Zbl 0572.46023 · doi:10.1007/BF01781073
[9] Bénilan, Ph, Boccardo, L., Gallouet, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An \[L^1\] L1-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations. Ann. Scuola Normale Superiore di Pisa, IV XXII, 241-273 (1995) · Zbl 0866.35037
[10] Berestycki, H., Hamel, F.: Front propagation in periodic excitable media. Comm. Pure Appl. Math. 55, 949-1032 (2002) · Zbl 1024.37054 · doi:10.1002/cpa.3022
[11] Berestycki, H., Hammel, F., Matano, H.: Bistable travelling waves around an obstacle. Comm. Pure Appl. Math. 62, 729-788 (2009) · Zbl 1172.35031 · doi:10.1002/cpa.20275
[12] Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. I - Periodic framework. J. Eur. Math. Soc. 7, 173-213 (2005) · Zbl 1142.35464 · doi:10.4171/JEMS/26
[13] Berestycki, H., Hamel, F., Nadirashvili, N.: The speed of propagation for KPP type problems. II: General domains. J. Am. Math. Soc. 23, 1-34 (2010) · Zbl 1197.35073 · doi:10.1090/S0894-0347-09-00633-X
[14] Berestycki, H., Nirenberg, L.: Travelling fronts in cylinders. Ann. Inst. H. Poincaré, Anal. Non Lin. 9, 497-572 (1992) · Zbl 0799.35073
[15] Brenier, Y.: Extended Monge-Kantorovich Theory. In: “Optimal Transportation and Applications”, Lectures given at the C.I.M.E. Summer School help in Martina Franca, L.A. Caffarelli and S. Salsa (eds.), Lecture Notes in Math. 1813, Springer-Verlag, 91-122 (2003)
[16] Cabré, X., Roquejoffre, J.-M.: Front propagation in Fisher-KPP equations with fractional diffusion. C. R. Math. Acad. Sci. Paris 347, 1361-1366 (2009) · Zbl 1182.35072 · doi:10.1016/j.crma.2009.10.012
[17] Caffarelli, L., Soria, F., Vázquez, J.L.: Regularity of solutions of the fractional porous medium. J. Eur. Math. Soc. (JEMS) 15, 1701-1746 (2013) · Zbl 1292.35312 · doi:10.4171/JEMS/401
[18] Constantin, A., Escher, J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559-568 (2011) · Zbl 1228.35076 · doi:10.4007/annals.2011.173.1.12
[19] Campos, J., Guerrero, P., Sánchez, O., Soler, J.: On the analysis of traveling waves to a nonlinear flux limited reaction-diffusion equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 141-155 (2013) · Zbl 1263.35059 · doi:10.1016/j.anihpc.2012.07.001
[20] Calvo, J., Mazón, J., Soler, J., Verbeni, M.: Qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens. Math. Mod. Meth. Appl. Sci. 21, 893-937 (2011) · Zbl 1223.35057 · doi:10.1142/S0218202511005416
[21] Caselles, V.: An existence and uniqueness result for flux limited diffusion equations. Discret. Continu. Dyn. Syst. 31, 1151-1195 (2011) · Zbl 1252.35163 · doi:10.3934/dcds.2011.31.1151
[22] Caselles, V.: On the entropy conditions for some flux limited diffusion equations. J. Diff. Eqs. 250, 3311-3348 (2011) · Zbl 1231.35101 · doi:10.1016/j.jde.2011.01.027
[23] Caselles, V.: Flux limited generalized porous media diffusion equations. Publicacions Matemàtiques 57, 155-217 (2013) · Zbl 1282.35205 · doi:10.5565/PUBLMAT_57113_07
[24] Caselles, V.: Convergence of flux limited porous media diffusion equations to its classical counterpart. Ann. Scuola Norm. Pisa XIV(2), 481-505 (2015) · Zbl 1433.35179
[25] Chen, G.Q., Frid, H.: Divergence-Measure Fields and Hyperbolic Conservation Laws. Arch. Ration. Mech. Anal. 147, 89-118 (1999) · Zbl 0942.35111 · doi:10.1007/s002050050146
[26] Chertock, A., Kurganov, A., Rosenau, P.: Formation of discontinuities in flux-saturated degenerate parabolic equations. Nonlinearity 16, 1875-1898 (2003) · Zbl 1049.35111 · doi:10.1088/0951-7715/16/6/301
[27] Dal, G.: Maso, Integral representation on \[BV(\Omega )\] BV(Ω) of \[\Gamma\] Γ-limits of variational integrals. Manuscr. Math. 30, 387-416 (1980) · Zbl 0435.49016
[28] De Cicco, V., Fusco, N., Verde, A.: On \[L^1\] L1-lower semicontinuity in \[BV\] BV. J. Convex Anal. 12, 173-185 (2005) · Zbl 1115.49011
[29] Fife, P.C., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335-361 (1977) · Zbl 0361.35035 · doi:10.1007/BF00250432
[30] Enguica, R., Gavioli, A., Sánchez, L.: A class of singular first order differential equations with applications in reaction-diffusion. Discret. Contin. Dyn. Syst. 33, 173-191 (2013) · Zbl 1269.34032 · doi:10.3934/dcds.2013.33.173
[31] Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugenics 7, 335-369 (1937) · JFM 63.1111.04
[32] Gatenby, R.A., Gawlinski, E.T.: A Reaction-Diffusion Model of Cancer Invasion. Cancer Res. 56, 5745-5753 (1996) · Zbl 1359.94032
[33] Jones, C.K.R.T., Gardner, R., Kapitula, T.: Stability of travelling waves for non-convex scalar viscous conservation laws. Comm. Pure Appl. Math. 46, 505-526 (1993) · Zbl 0791.35078 · doi:10.1002/cpa.3160460404
[34] Hadeler, K.P., Rothe, F.: Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251-263 (1975) · Zbl 0343.92009 · doi:10.1007/BF00277154
[35] Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964) · Zbl 0125.32102
[36] Kolmogorov, N., Petrovsky, I.G., Piskunov, N.S.: Étude de l’équation de la diffusion avec croissance de la quantité de matiére et son application à un probléme biologique, Bulletin Université d’Etat à Moscou (Bjul. Moskowskogo Gos. Univ.), Série internationale A 1, 1-26 (1937) · Zbl 0435.49016
[37] Kondo, S., Miura, T.: Reaction-Diffusion model as a framework for understanding biological pattern formation. Science 329, 1616-1620 (2010) · Zbl 1226.35077 · doi:10.1126/science.1179047
[38] Kruzhkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR-Sb. 10, 217-243 (1970) · Zbl 0215.16203 · doi:10.1070/SM1970v010n02ABEH002156
[39] Kurganov, A., Rosenau, P.: Effects of a saturating dissipation in Burgers-type equations. Commun. Pure Appl. Math. 50, 753-771 (1997) · Zbl 0888.35097 · doi:10.1002/(SICI)1097-0312(199708)50:8<753::AID-CPA2>3.0.CO;2-5
[40] Kurganov, A., Levy, D., Rosenau, P.: On Burgers-type equations with nonmonotonic dissipative fluxes. Commun. Pure Appl. Math. 51, 443-473 (1998) · Zbl 0929.35138 · doi:10.1002/(SICI)1097-0312(199805)51:5<443::AID-CPA1>3.0.CO;2-8
[41] Kurganov, A., Rosenau, P.: On reaction processes with saturating diffusion. Nonlinearity 19, 171-193 (2006) · Zbl 1094.35063 · doi:10.1088/0951-7715/19/1/009
[42] Majda, A.J., Souganidis, P.E.: Flame fronts in a turbulent combustion model with fractal velocity fields. Comm. Pure Appl. Math. 51, 1337-1348 (1998) · Zbl 0939.35097 · doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1337::AID-CPA4>3.0.CO;2-B
[43] Marquina, A.: Diffusion front capturing schemes for a class of FokkerPlanck equations: Application to the relativistic heat equation. J. Comput. Phys. 229, 2659-2674 (2010) · Zbl 1423.35365 · doi:10.1016/j.jcp.2009.12.014
[44] McCann, R., Puel, M.: Constructing a relativistic heat flow by transport time steps. Ann. Inst. H. Poincaré Anal. Non Lineaire 26, 2539-2580 (2009) · Zbl 1181.80001 · doi:10.1016/j.anihpc.2009.06.006
[45] Meinhardt, H., Prusinkiewicz, P., Fowler, D.R.: The algorithmic beauty of sea shells. Springer-Verlag, NY (1998) · doi:10.1007/978-3-662-03617-4
[46] Murray, J.D.: Mathematical Biology. Springer-Verlag, NY (1996) · Zbl 0704.92001
[47] Mueller, C., Mytnik, L., Quastel, J.: Effect of noise on front propagation in reaction-diffusion equations of KPP type. Inv. Math. 184, 405-453 (2011) · Zbl 1222.35105 · doi:10.1007/s00222-010-0292-5
[48] Newman, W.I.: Some Exact Solutions to a Non-linear Diffusion Problem in Population Genetics and Combustion. J. Theor. Biol 85, 325-334 (1980) · doi:10.1016/0022-5193(80)90024-7
[49] Newman, W.I., Sagan, C.: Galactic Civilizations: Populations Dynamics and Interstellar Diffusion. Icarus 46, 293-327 (1981) · doi:10.1016/0019-1035(81)90135-4
[50] Ngamsaad, W., Khomphurngson, K.: Self-similar solutions to a density-dependent reaction-diffusion model. Phys. Rev. E 85, 066120 (2012) · doi:10.1103/PhysRevE.85.066120
[51] de Pablo, A., Vázquez, J.L.: Travelling waves and finite propagation in a reaction-diffusion equation. J. Differ. Equ. 93, 19-61 (1991) · Zbl 0784.35045 · doi:10.1016/0022-0396(91)90021-Z
[52] de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A general fractional porous medium equation. Comm. Pure Appl. Math. 65, 1242-1284 (2012) · Zbl 1248.35220 · doi:10.1002/cpa.21408
[53] Rosenau, P.: Tempered Diffusion: A Transport Process with Propagating Front and Inertial Delay. Phys. Rev. A 46, 7371-7374 (1992) · doi:10.1103/PhysRevA.46.R7371
[54] Rosenau, P.: Reaction and concentration dependent diffusion model. Phys. Rev. Lett. 88, 194501 (2002) · doi:10.1103/PhysRevLett.88.194501
[55] Sánchez-Garduño, F., Maini, P.K.: Existence and uniqueness of a sharp traveling wave in degenerate non-linear diffusion Fisher-KPP equations. J. Math. Biol. 33, 163-192 (1994) · Zbl 0822.92021 · doi:10.1007/BF00160178
[56] Sánchez-Garduño, F., Maini, P.K.: Traveling wave phenomena in Some Degenerate Reaction-Diffusion Equations. J. Diff. Eq. 177, 281-319 (1995) · Zbl 0821.35085 · doi:10.1006/jdeq.1995.1055
[57] Sánchez-Garduño, F., Maini, P.K., Kappos, M.E.: A shooting argument approach to a sharp-type solution for nonlinear degenerate Fisher-KPP equations. IMA J. Appl. Math. 57, 211-221 (1996) · Zbl 0876.35066 · doi:10.1093/imamat/57.3.211
[58] Vázquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Univ. Press, Oxford (2006) · Zbl 1113.35004 · doi:10.1093/acprof:oso/9780198569039.001.0001
[59] Verbeni, M., Sánchez, O., Mollica, E., Siegl-Cachedenier, I., Carleton, A., Guerrero, I., Ruiz i Altaba, A., Soler, J.: Modeling morphogenetic action through flux-limited spreading. Phys. Life Rev. 10, 457-475 (2013) · doi:10.1016/j.plrev.2013.06.004
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.