We will describe some existence results for the mean field games PDE system with nonseparable Hamiltonian. Initial data will be taken in Sobolev spaces. Since little will be assumed about the nature of the Hamiltonian, smallness conditions will need to be taken. The payoff function can either be taken to be smoothing or nonsmoothing; if it is not smoothing, then a further smallness condition is taken. We will also briefly discuss a specific mean field games system with nonseparable Hamiltonian, which is a model for household savings and wealth, giving a nonexistence result for this system.
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