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In this paper, we investigate the Sobolev regularity for mean-field games inthe whole space $\Rr^d$. This is achieved by combining integrability for thesolutions of the Fokker-Planck equation with estimates for the Hamilton-Jacobiequation in Sobolev spaces. To avoid the mathematical challenges posed due tothe lack of compactness, we prove an entropy dissipation estimate for theadjoint variable. This, together with the non-linear adjoint method, yieldsuniform estimates for solutions of the Hamilton-Jacobi equation in$W^{1,p}_{loc}(\Rr^d)$.

Local regularity for mean-field games in the whole space