Nonlinear regularization estimates and global well-posedness for the Landau-Coulomb equation near equilibrium

Abstract

We consider the Landau equation with Coulomb potential in the spatially homogeneous case. We show short time propagation of smallness in $L^p$ norms for $p>3/2$ and instantaneous regularization in Sobolev spaces. This yields new short time quantitative a priori estimates that are unconditional near equilibrium. We combine these estimates with existing literature on global well-posedness for regular data to extend the well-posedness theory to small $L^p$ data with $p$ arbitrarily close to $3/2$. The threshold $p = 3/2$ agrees with previous work on conditional regularity for the Landau equation in the far from equilibrium regime. In light of the monotonicity of the Fisher information shown in the recent preprint [arXiv:2311.09420], our primary nonlinear regularization estimate holds even in the far-from-equilibrium regime. As a consequence, we obtain exponential convergence to equilibrium for suitably localized solutions in every Sobolev norm.