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This work deals with the Landau equation for very soft and Coulomb potentials near the associated Maxwellian equilibrium. We first investigate the corresponding linearized operator and develop a method to prove strong asymptotical (but not uniformly exponential) stability estimates of its associated semigroup in large functional spaces. We then deduce existence, uniqueness and fast decay of the solutions to the nonlinear equation in a close-to-equilibrium framework. Our result drastically improves the set of initial data compared to the one considered by Guo and Strain who established similar results in Guo (Commun Math Phys 231:391–434, 2002) and Strain and Guo (Commun Partial Differ Equ 31(1–3):417–429, 2006; Arch Ration Mech Anal 187(2):287–339, 2008). Our functional framework is compatible with the non perturbative frameworks developed by Villani (Arch Ration Mech Anal 143(3):273–307 1998), Desvillettes and Villani (Invent Math 159(2):245–316, 2005), Desvillettes (J Funct Anal 269(5):1359–1403, 2015) and Carrapatoso et al. (arXiv:1510.08704, 2016), and our main result then makes possible to improve the speed of convergence to the equilibrium established therein.

Landau Equation for Very Soft and Coulomb Potentials Near Maxwellians