KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
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AbstractIn this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity.
CitationTime-Dependent Mean-Field Games with Logarithmic Nonlinearities 2015, 47 (5):3798 SIAM Journal on Mathematical Analysis