American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals
  • DCDS Home
  • This Issue
  • Next Article
    Subshifts of finite type which have completely positive entropy
October  2011, 29(4): 1517-1552. doi: 10.3934/dcds.2011.29.1517

Repeated games for non-linear parabolic integro-differential equations and integral curvature flows

Cyril Imbert 1, and  Sylvia Serfaty 2,

1. 

CEREMADE, UMR CNRS 7534, université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16
2. 

UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France

Received  January 2010 Revised  September 2010 Published  December 2010

The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations.
   In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For parabolic integro-differential equations, players choose smooth functions on the whole space. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces.
Keywords: geometric flows., viscosity solutions, parabolic integro-differential equations, Repeated games, integral curvature flows.
Mathematics Subject Classification: 35C99, 53C44, 90D10, 49K2.
Citation: Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517
References:
[1]

N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data,, Trans. Amer. Math. Soc., 361 (2009), 2527.  doi: 10.1090/S0002-9947-08-04758-2.  Google Scholar

[2]

O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution,, Arch. Ration. Mech. Anal., 181 (2006), 449.  doi: 10.1007/s00205-006-0418-5.  Google Scholar

[3]

G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited,, Annales de l'Institut Henri Poincaré, 25 (2008), 567.  doi: 10.1016/j.anihpc.2007.02.007.  Google Scholar

[4]

G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory,, SIAM J. Control Optim., 31 (1993), 439.  doi: 10.1137/0331021.  Google Scholar

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Anal., 4 (1991), 271.   Google Scholar

[6]

L. Caffarelli, J.-M. Roquejoffre and O. Savin, Non local minimal surfaces,, 2009, ().   Google Scholar

[7]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar

[8]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Rational Mech. Anal., 180 (2010), 301.   Google Scholar

[9]

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, J. Differential Geom., 33 (1991), 749.   Google Scholar

[10]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Financial Mathematics Series, (2004).   Google Scholar

[11]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1.   Google Scholar

[12]

M. G. Crandall and P.-L. Lions, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre,, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 183.   Google Scholar

[13]

F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics,, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061.  doi: 10.4171/JEMS/140.  Google Scholar

[14]

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations,, Indiana Univ. Math. J., 33 (1984), 773.  doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[15]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, J. Differential Geom., 33 (1991), 635.   Google Scholar

[16]

N. Forcadel, C. Imbert, and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics,, Discrete Contin. Dyn. Syst., 23 (2009), 785.  doi: 10.3934/dcds.2009.23.785.  Google Scholar

[17]

C. Imbert, Level set approach for fractional mean curvature flows,, Interfaces Free Bound., 11 (2009), 153.  doi: 10.4171/IFB/207.  Google Scholar

[18]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications,, preprint, (2009).   Google Scholar

[19]

E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations,, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 137.  doi: 10.1007/s00030-005-0031-6.  Google Scholar

[20]

R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344.  doi: 10.1002/cpa.20101.  Google Scholar

[21]

R. V. Kohn and S. Serfaty, A deterministic-control-based approach to fully non-linear parabolic and elliptic equations,, Comm. Pure Appl. Math., 63 (2010), 1298.  doi: 10.1002/cpa.20336.  Google Scholar

[22]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", vol. 69 of Research Notes in Mathematics, 69 (1982).   Google Scholar

[23]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[24]

A. Sayah, Équations de Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité,, Comm. Partial Differential Equations, 16 (1991), 1057.   Google Scholar

[25]

H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions,, in, 10 (1988), 501.   Google Scholar

[26]

P. E. Souganidis, Front propagation: Theory and applications,, in, 1660 (1997), 186.   Google Scholar

[27]

J. Spencer, Balancing games,, J. Combinatorial Theory Ser. B, 23 (1977), 68.  doi: 10.1016/0095-8956(77)90057-0.  Google Scholar

show all references

References:
[1]

N. Alibaud and C. Imbert, Fractional semi-linear parabolic equations with unbounded data,, Trans. Amer. Math. Soc., 361 (2009), 2527.  doi: 10.1090/S0002-9947-08-04758-2.  Google Scholar

[2]

O. Alvarez, P. Hoch, Y. Le Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution,, Arch. Ration. Mech. Anal., 181 (2006), 449.  doi: 10.1007/s00205-006-0418-5.  Google Scholar

[3]

G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited,, Annales de l'Institut Henri Poincaré, 25 (2008), 567.  doi: 10.1016/j.anihpc.2007.02.007.  Google Scholar

[4]

G. Barles, H. M. Soner and P. E. Souganidis, Front propagation and phase field theory,, SIAM J. Control Optim., 31 (1993), 439.  doi: 10.1137/0331021.  Google Scholar

[5]

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations,, Asymptotic Anal., 4 (1991), 271.   Google Scholar

[6]

L. Caffarelli, J.-M. Roquejoffre and O. Savin, Non local minimal surfaces,, 2009, ().   Google Scholar

[7]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar

[8]

L. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation,, Arch. Rational Mech. Anal., 180 (2010), 301.   Google Scholar

[9]

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, J. Differential Geom., 33 (1991), 749.   Google Scholar

[10]

R. Cont and P. Tankov, "Financial Modelling with Jump Processes,", Financial Mathematics Series, (2004).   Google Scholar

[11]

M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1.   Google Scholar

[12]

M. G. Crandall and P.-L. Lions, Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre,, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 183.   Google Scholar

[13]

F. Da Lio, N. Forcadel and R. Monneau, Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocation dynamics,, J. Eur. Math. Soc. (JEMS), 10 (2008), 1061.  doi: 10.4171/JEMS/140.  Google Scholar

[14]

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations,, Indiana Univ. Math. J., 33 (1984), 773.  doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[15]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I,, J. Differential Geom., 33 (1991), 635.   Google Scholar

[16]

N. Forcadel, C. Imbert, and R. Monneau, Homogenization of some particle systems with two-body interactions and of the dislocation dynamics,, Discrete Contin. Dyn. Syst., 23 (2009), 785.  doi: 10.3934/dcds.2009.23.785.  Google Scholar

[17]

C. Imbert, Level set approach for fractional mean curvature flows,, Interfaces Free Bound., 11 (2009), 153.  doi: 10.4171/IFB/207.  Google Scholar

[18]

C. Imbert and P. E. Souganidis, Phasefield theory for fractional reaction-diffusion equations and applications,, preprint, (2009).   Google Scholar

[19]

E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations,, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 137.  doi: 10.1007/s00030-005-0031-6.  Google Scholar

[20]

R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature,, Comm. Pure Appl. Math., 59 (2006), 344.  doi: 10.1002/cpa.20101.  Google Scholar

[21]

R. V. Kohn and S. Serfaty, A deterministic-control-based approach to fully non-linear parabolic and elliptic equations,, Comm. Pure Appl. Math., 63 (2010), 1298.  doi: 10.1002/cpa.20336.  Google Scholar

[22]

P.-L. Lions, "Generalized Solutions of Hamilton-Jacobi Equations,", vol. 69 of Research Notes in Mathematics, 69 (1982).   Google Scholar

[23]

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,, J. Comput. Phys., 79 (1988), 12.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[24]

A. Sayah, Équations de Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité,, Comm. Partial Differential Equations, 16 (1991), 1057.   Google Scholar

[25]

H. M. Soner, Optimal control of jump-Markov processes and viscosity solutions,, in, 10 (1988), 501.   Google Scholar

[26]

P. E. Souganidis, Front propagation: Theory and applications,, in, 1660 (1997), 186.   Google Scholar

[27]

J. Spencer, Balancing games,, J. Combinatorial Theory Ser. B, 23 (1977), 68.  doi: 10.1016/0095-8956(77)90057-0.  Google Scholar

[1]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[2]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[3]

Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020109
[4]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268
[5]

Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313
[6]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
[7]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[8]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328
[9]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[10]

Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49
[11]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265
[12]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455
[13]

Nicholas Geneva, Nicholas Zabaras. Multi-fidelity generative deep learning turbulent flows. Foundations of Data Science, 2020, 2 (4) : 391-428. doi: 10.3934/fods.2020019
[14]

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020280
[15]

Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398
[16]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318
[17]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122
[18]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189
[19]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161
[20]

Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021009

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences