In this talk, we study the spatially homogeneous Landau equation. The equation can be understood as a nonlinear heat equation with nonlocal coefficients. The equation can be obtained from the classical Boltzmann equation in the grazing collision limit. For the hard potentials, I will introduce a proof of the existence of axi-symmetric measure-valued solutions for any axi-symmetric $\mathcal{P}_2(\mathbb{R}^3)$ initial profile. Moreover, we will see that if the initial data is not a single Dirac mass, then the solution instantaneously becomes analytic for any time $t>0.$