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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] J. Hulshof, Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl., 157 (1991), 75-111. Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. Google Scholar [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. Google Scholar [17] J. L. Vázquez, "The Porous Medium Equation," Mathematical Theory, Oxford Mathematical Monographs, Oxford, 2007. Google Scholar [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. Google Scholar [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. Google Scholar [20] C. Villani, "Optimal Transport, Old and New," Springer-Verlag, 2008. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] J. Hulshof, Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl., 157 (1991), 75-111. Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [15] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174. Google Scholar [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. Google Scholar [17] J. L. Vázquez, "The Porous Medium Equation," Mathematical Theory, Oxford Mathematical Monographs, Oxford, 2007. Google Scholar [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. Google Scholar [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. Google Scholar [20] C. Villani, "Optimal Transport, Old and New," Springer-Verlag, 2008. Google Scholar

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