no. 4
On the minimizing movement with the 1-Wasserstein distance
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1415-1427.

We consider a class of doubly nonlinear constrained evolution equations which may be viewed as a nonlinear extension of the growing sandpile model of [L. Prigozhin, Eur. J. Appl. Math.  7 (1996) 225–235.]. We prove existence of weak solutions for quite irregular sources by a semi-implicit scheme in the spirit of the seminal works of [R. Jordan et al., SIAM J. Math. Anal.  29 (1998) 1–17, D. Kinderlehrer and N.J. Walkington, Math. Model. Numer. Anal.  33 (1999) 837–852.] but with the 1-Wasserstein distance instead of the quadratic one. We also prove an L1-contraction result when the source is L1 and deduce uniqueness and stability in this case.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017055
Classification : 35K55, 35D30, 49N15
Mots clés : 1-Wasserstein distance, minimizing movement, L1-contraction, growing sandpiles
Agueh, Martial 1 ; Carlier, Guillaume 1 ; Igbida, Noureddine 1

1 
@article{COCV_2018__24_4_1415_0,
     author = {Agueh, Martial and Carlier, Guillaume and Igbida, Noureddine},
     title = {On the minimizing movement with the {1-Wasserstein} distance},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1415--1427},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {4},
     year = {2018},
     doi = {10.1051/cocv/2017055},
     zbl = {1414.35109},
     mrnumber = {3922439},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2017055/}
}
TY  - JOUR
AU  - Agueh, Martial
AU  - Carlier, Guillaume
AU  - Igbida, Noureddine
TI  - On the minimizing movement with the 1-Wasserstein distance
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 1415
EP  - 1427
VL  - 24
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2017055/
DO  - 10.1051/cocv/2017055
LA  - en
ID  - COCV_2018__24_4_1415_0
ER  - 
%0 Journal Article
%A Agueh, Martial
%A Carlier, Guillaume
%A Igbida, Noureddine
%T On the minimizing movement with the 1-Wasserstein distance
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 1415-1427
%V 24
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2017055/
%R 10.1051/cocv/2017055
%G en
%F COCV_2018__24_4_1415_0
Agueh, Martial; Carlier, Guillaume; Igbida, Noureddine. On the minimizing movement with the 1-Wasserstein distance. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1415-1427. doi : 10.1051/cocv/2017055. http://www.numdam.org/articles/10.1051/cocv/2017055/

[1] G. Aronsson and L.C. Evans, An asymptotic model for compression molding. Indiana Univ. Math. J.  51 (2002) 1–36 | DOI | MR | Zbl

[2] G. Aronsson, L.C. Evans and Y. Wu, Fast/slow diffusion and growing sandpiles. J. Diff. Equ.  131 (1996) 304–335 | DOI | MR | Zbl

[3] J.-P. Aubin, Un théorème de compacité. C.R. Acad. Sci. Paris 256 (1963) 5042–5044 | MR | Zbl

[4] H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise.  Masson, Paris (1983) | MR | Zbl

[5] P. Cannarsa and P. Cardaliaguet, Representation of equilibrium solutions to the table problem for growing sandpiles. J. Eur. Math. Soc. (JEMS) 6 (2004) 435–464 | DOI | MR | Zbl

[6] P. Cannarsa, P. Cardaliaguet and C. Sinestrari, On a differential model for growing sandpiles with non-regular sources. Comm. Partial Diff. Equ.  34 (2009) 656–675 | DOI | MR | Zbl

[7] J. Carillo, Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal.  147 (1999) 269–361 | DOI | MR | Zbl

[8] M.G. Crandall and T.M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces. Amer. J. Math.  93 (1971) 265–298 | DOI | MR | Zbl

[9] L. De Pascale and C. Jimenez, Duality theory and optimal transport for sand piles growing in a silos. Adv. Diff. Equ.  20 (2015) 859–886 | MR | Zbl

[10] S. Dumont and N. Igbida, On a dual formulation for the growing sandpile problem. Eur. J. Appl. Math.  20 (2009) 169–185 | DOI | MR | Zbl

[11] L.C. Evans, M. Feldman and R.F. Gariepy, Fast/slow diffusion and collapsing sandpiles. J. Diff. Equ.  137 (1997) 166–209 | DOI | MR | Zbl

[12] N. Igbida, Evolution monge-kantorovich equation. J. Diff. Equ.  225 (2013) 1383–1407 | DOI | MR | Zbl

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

[14] D. Kinderlehrer and N.J. Walkington, Approximation of parabolic equations using the Wasserstein metric. Math. Model. Numer. Anal.  33 (1999) 837–852 | DOI | Numdam | MR | Zbl

[15] L. Prigozhin, Variational model of sandpile growth. Eur. J. Appl. Math.  7 (1996) 225–235 | DOI | MR | Zbl

[16] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl.  146 (1987) 65–96 | DOI | MR | Zbl

[17] C. Villani, Topics in Optimal Transportation. In Vol. 58 of Graduate Studies in Mathematics.  American Mathematical Society, Providence (2003) | MR | Zbl

Cité par Sources :