# American Institute of Mathematical Sciences

September  2012, 7(3): 525-541. doi: 10.3934/nhm.2012.7.525

## On the signed porous medium flow

 1 Département de Mathématiques, UMR 8628 Université Paris-Sud 11-CNRS, Bâtiment 425, Faculté des Sciences d'Orsay, Université Paris-Sud 11, F-91405 Orsay Cedex, France

Received  December 2011 Revised  July 2012 Published  October 2012

We prove that the signed porous medium equation can be regarded as limit of an optimal transport variational scheme, therefore extending the classical result for positive solutions of [13] and showing that an optimal transport approach is suited even for treating signed densities.
Citation: Edoardo Mainini. On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7 (3) : 525-541. doi: 10.3934/nhm.2012.7.525
##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Spaces of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [2] L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217-246. [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (eds. A. Fasano and M. Primicerio), Lecture Notes in Math. 1224, Springer, Berlin, (1986), 1-46. [4] M. Bertsch and D. Hilhorst, The interface between regions where $u<0$ and $u>0$ in the porous medium equation, Appl. Anal., 41 (1991), 111-130. [5] J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263. [6] A. Friedman and S. Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. [7] J. Hulshof, Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl., 157 (1991), 75-111. [8] J. Hulshof, J. R. King and M. Bowen, Intermediate asymptotics of the porous medium equation with sign changes, Adv. Differential Equations, 6 (2001), 1115-1152. [9] J. Hulshof and J. L. Vázquez, The dipole solution for the porous medium equation in several space dimensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20 (1993), 193-217. [10] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [11] S. Kamin and J. L. Vázquez, Asymptotic behaviour of solutions of the porous medium equation with changing sign, SIAM J. Math. Anal., 22 (1991), 34-45. [12] E. Mainini, A description of transport cost for signed measures, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 390 (2011), 147-181. [13] F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103. doi: 10.1007/s002050050073. [14] F. Otto, Evolution of microstructure in unstable porous media flow: A relaxational approach, Comm. Pure Appl. Math., 52 (1999), 873-915. doi: 10.1002/(SICI)1097-0312(199907)52:7<873::AID-CPA5>3.0.CO;2-T. [15] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. [16] C. J. van Duijn, S. M. Gomes and H. F. Zhang, On a class of similarity solutions of the equation $u_t=(|u|^{m-1} u_x)_x$ with $m > -1$, IMA J. Appl. Math., 41 (1988), 147-163. [17] J. L. Vázquez, "The Porous Medium Equation," Mathematical Theory, Oxford Mathematical Monographs, Oxford, 2007. [18] J. L. Vázquez, Asymptotic beahviour for the porous medium equation posed in the whole space, Dedicated to Philippe Bénilan. J. Evol. Equ., 3 (2003), 67-118. [19] J. L. Vázquez, New self-similar solutions of the porous medium equation and the theory of solutions of changing sign, Nonlinear Anal., 15 (1990), 931-942. [20] C. Villani, "Optimal Transport, Old and New," Springer-Verlag, 2008.

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##### References:
 [1] L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Spaces of Probability Measures," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. [2] L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217-246. [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (eds. A. Fasano and M. Primicerio), Lecture Notes in Math. 1224, Springer, Berlin, (1986), 1-46. [4] M. Bertsch and D. Hilhorst, The interface between regions where $u<0$ and $u>0$ in the porous medium equation, Appl. Anal., 41 (1991), 111-130. [5] J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263. [6] A. Friedman and S. Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563. [7] J. Hulshof, Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl., 157 (1991), 75-111. [8] J. Hulshof, J. R. King and M. Bowen, Intermediate asymptotics of the porous medium equation with sign changes, Adv. Differential Equations, 6 (2001), 1115-1152. [9] J. Hulshof and J. L. Vázquez, The dipole solution for the porous medium equation in several space dimensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20 (1993), 193-217. [10] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359. [11] S. Kamin and J. L. Vázquez, Asymptotic behaviour of solutions of the porous medium equation with changing sign, SIAM J. Math. Anal., 22 (1991), 34-45. [12] E. Mainini, A description of transport cost for signed measures, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 390 (2011), 147-181. [13] F. Otto, Dynamics of labyrinthine pattern formation in magnetic fluids: A mean-field theory, Arch. Rational Mech. Anal., 141 (1998), 63-103. doi: 10.1007/s002050050073. [14] F. Otto, Evolution of microstructure in unstable porous media flow: A relaxational approach, Comm. Pure Appl. Math., 52 (1999), 873-915. doi: 10.1002/(SICI)1097-0312(199907)52:7<873::AID-CPA5>3.0.CO;2-T. [15] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. [16] C. J. van Duijn, S. M. Gomes and H. F. Zhang, On a class of similarity solutions of the equation $u_t=(|u|^{m-1} u_x)_x$ with $m > -1$, IMA J. Appl. Math., 41 (1988), 147-163. [17] J. L. Vázquez, "The Porous Medium Equation," Mathematical Theory, Oxford Mathematical Monographs, Oxford, 2007. [18] J. L. Vázquez, Asymptotic beahviour for the porous medium equation posed in the whole space, Dedicated to Philippe Bénilan. J. Evol. Equ., 3 (2003), 67-118. [19] J. L. Vázquez, New self-similar solutions of the porous medium equation and the theory of solutions of changing sign, Nonlinear Anal., 15 (1990), 931-942. [20] C. Villani, "Optimal Transport, Old and New," Springer-Verlag, 2008.
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