A variational method for a class of parabolic PDEs
Figalli, Alessio1 ; Gangbo, Wilfrid2 ; Yolcu, Türkay3
1 Department of Mathematics The University of Texas at Austin Austin TX, USA
2 Georgia Institute of Technology Atlanta GA, USA
3 Department of Mathematics Purdue University West Lafayette, IN, USA
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 1, pp. 207-252.

In this manuscript we extend De Giorgi’s interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but it does not induce a metric. Assuming the initial condition to be a density function, not necessarily smooth, but solely of bounded first moments and finite “entropy”, we use a variational scheme to discretize the equation in time and construct approximate solutions. Then De Giorgi’s interpolation method is revealed to be a powerful tool for proving convergence of our algorithm. Finally we show uniqueness and stability in L 1 of our solutions.

Publié le :
Classification : 35K59, 49J40, 82C40, 47J25
Figalli, Alessio 1 ; Gangbo, Wilfrid 2 ; Yolcu, Türkay 3

1 Department of Mathematics The University of Texas at Austin Austin TX, USA
2 Georgia Institute of Technology Atlanta GA, USA
3 Department of Mathematics Purdue University West Lafayette, IN, USA 
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     title = {A variational method for a class of parabolic {PDEs}},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     publisher = {Scuola Normale Superiore, Pisa},
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Figalli, Alessio; Gangbo, Wilfrid; Yolcu, Türkay. A variational method for a class of parabolic PDEs. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 1, pp. 207-252. http://www.numdam.org/item/ASNSP_2011_5_10_1_207_0/

[1] M.  Agueh, Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory, Adv. Differential Equations 10 (2005), 309–360. | MR | Zbl

[2] H-W.  Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. | EuDML | MR | Zbl

[3] L. Ambrosio, N. Gigli and G. Savaré, “Gradient Flows in Metric Spaces and the Wasserstein Spaces of Probability Measures”, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005. | MR | Zbl

[4] P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. 9 (2007), 85–121. | EuDML | MR | Zbl

[5] B. Dacorogna, “Direct Methods in the Calculus of Variations”, Springer-Verlag, Berlin, 1989. | MR | Zbl

[6] E. De Giorgi, New problems on minimizing movements, In: “Boundary Value Problems for PDE and Applications”, C. Baiocchi and J. L. Lions (eds.), Masson, 1993, 81–98. | MR | Zbl

[7] E. De Giorgi, A. Mariono and M. Tosques, Problems of evolutions in metric spaces and maximal decreasing curve, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Mat. Natur. (8) 68 (1980), 180–187. | EuDML | MR | Zbl

[8] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds, Israel J. Math. 175 (2010), 1–59. | MR | Zbl

[9] A. Figalli, Existence, uniqueness, and regularity of optimal transport maps, SIAM J. Math. Anal. 39 (2007) 126–137. | MR | Zbl

[10] W. Gangbo, Quelques problèmes d’analyse convexe, Rapport d’habilitation à diriger des recherches, Jan. 1995. Available at http://www.math.gatech.edu/ gangbo/publications/.

[11] W. Gangbo and R. McCann, Optimal maps in Monge’s mass transport problem, C. R. Acad. Sci. Paris Série I Math. 321 (1995), 1653–1658. | MR | Zbl

[12] W. Gangbo and R. McCann, The geometry of optimal transportation, Acta Math. 177 (1996), 113–161. | MR | Zbl

[13] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math Anal. 29 (1998), 1–17. | MR | Zbl

[14] S. Lisini, Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces, ESAIM Control Optim. Calc. Var. 15 (2009), 712–740. | EuDML | Numdam | MR | Zbl

[15] F. Otto, L 1 -contraction and uniqueness for quasilinear elliptic–parabolic equations, J. Differential Equations 131 (1996), 20–38. | MR | Zbl

[16] C. Villani, “Topics in Optimal Transportation”, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, 2003. | MR | Zbl

[17] C. Villani, “Optimal Transport, Old and New”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 338, Springer-Verlag, Berlin, 2009. | MR | Zbl

[18] T. Yolcu, “Parabolic Systems with an Underlying Lagrangian”, PhD Dissertation, Georgia Tech., 2009. | MR