Kinetictype Mean Field Games with Nonseparable Local Hamiltonians
Abstract
We prove wellposedness of a class of kinetictype Mean Field Games, which typically arise when agents control their acceleration. Such systems include independent variables representing the spatial position as well as velocity. We consider nonseparable Hamiltonians without any structural conditions, which depend locally on the density variable. Our analysis is based on two main ingredients: an energy method for the forwardbackward system in Sobolev spaces, on the one hand and on a suitable vector field method to control derivatives with respect to the velocity variable, on the other hand. The careful combination of these two techniques reveals interesting phenomena applicable for Mean Field Games involving general classes of driftdiffusion operators and nonlinearities. While many prior existence theories for general Mean Field Games systems take the final datum function to be smoothing, we can allow this function to be nonsmoothing, i.e. also depending locally on the final measure. Our wellposedness results hold under an appropriate smallness condition, assumed jointly on the data.
 Publication:

arXiv eprints
 Pub Date:
 March 2024
 DOI:
 10.48550/arXiv.2403.12829
 arXiv:
 arXiv:2403.12829
 Bibcode:
 2024arXiv240312829A
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Optimization and Control
 EPrint:
 26 pages