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