On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials

We investigate in this work the rate of convergence to equilibrium of solutions to the spatially homogeneous Landau equation with soft potentials. Firstly, we prove a polynomial in time convergence using an entropy method with some new a priori estimates. Finally, we prove an exponential in time convergence towards the equilibrium with the optimal rate, given by the spectral gap of the associated linearised operator, combining new decay estimates for the semigroup generated by the linearised Landau operator in weighted L^p-spaces together with the polynomial decay described above.