For mean field games with local coupling, existence results are typically for weak solutions rather than strong solutions. We identify conditions on the Hamiltonian and the coupling, which allow us to prove the existence of small, locally unique, strong solutions over any finite time interval in the case of local coupling; these conditions place us in the case of superquadratic Hamiltonians. For the regularity of solutions, we find that at each time in the interior of the time interval, the Fourier coefficients of the solutions decay exponentially. The method of proof is inspired by the work of Duchon and Robert on vortex sheets in incompressible fluids.
Pour les jeux à champ moyens avec couplage local, les résultats d'existence sont typiquement obtenus pour des solutions faibles plutôt que pour des solutions fortes. Nous identifions des conditions sur le Hamiltonien et sur le couplage qui nous permettent de démontrer l'existence d'une solution forte, petite et localement unique pour tout intervalle de temps fini dans le cas d'un couplage local ; ces conditions nous placent dans une situation de Hamiltonien super-quadratique. Pour la régularité des solutions, nous trouvons que, pour chaque point dans l'intérieur de l'intervalle de temps, les coefficients de Fourier des solutions décroissent exponentiellement. La preuve est inspirée par les travaux de Duchon et Robert sur les nappes de tourbillons de fluides incompressibles.
Accepted:
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David M. Ambrose 1
@article{CRMATH_2016__354_6_589_0, author = {David M. Ambrose}, title = {Small strong solutions for time-dependent mean field games with local coupling}, journal = {Comptes Rendus. Math\'ematique}, pages = {589--594}, publisher = {Elsevier}, volume = {354}, number = {6}, year = {2016}, doi = {10.1016/j.crma.2016.02.006}, language = {en}, }
David M. Ambrose. Small strong solutions for time-dependent mean field games with local coupling. Comptes Rendus. Mathématique, Volume 354 (2016) no. 6, pp. 589-594. doi : 10.1016/j.crma.2016.02.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.006/
[1] Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci., Volume 12 (2002) no. 4, pp. 283-318
[2] Singular solutions and ill-posedness for the evolution of vortex sheets, SIAM J. Math. Anal., Volume 20 (1989) no. 2, pp. 293-307
[3] Global vortex sheet solutions of Euler equations in the plane, J. Differ. Equ., Volume 73 (1988) no. 2, pp. 215-224
[4] Time-dependent mean-field games in the superquadratic case, 2014 | arXiv
[5] Time-dependent mean-field games in the subquadratic case, Commun. Partial Differ. Equ., Volume 40 (2015), pp. 40-76
[6] Mean field games and applications, Paris–Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., vol. 2003, Springer, Berlin, 2011, pp. 205-266
[7] Jeux à champ moyen I : Le cas stationnaire, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 619-625
[8] Jeux à champ moyen II : Horizin fini et contrôle optimal, C. R. Acad. Sci. Paris, Ser. I, Volume 343 (2006), pp. 679-684
[9] Mean field games, Jpn. J. Math., Volume 2 (2007) no. 1, pp. 229-260
[10] Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d, ESAIM Control Optim. Calc. Var., Volume 8 (2002), pp. 801-825 (electronic). A tribute to J.-L. Lions
[11] Temporal boundary value problems in interfacial fluid dynamics, Appl. Anal., Volume 92 (2013) no. 5, pp. 922-948
[12] Weak solutions to Fokker–Planck equations and mean field games, Arch. Ration. Mech. Anal., Volume 216 (2015), pp. 1-62
[13] Mathematical analysis of vortex sheets, Commun. Pure Appl. Math., Volume 59 (2006) no. 8, pp. 1065-1206
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