A blow-down mechanism for the Landau-Coulomb equation

We investigate the Landau-Coulomb equation and show an explicit blow-down mechanism for a family of initial data that are small-scale, supercritical perturbations of a Maxwellian function. We establish global well-posedness and show that the initial bump region will disappear in a time of order one. We prove that the function remains close to an explicit function during the blow-down. As a consequence, our result shows exponential decay in time of the solution towards equilibrium. The key ingredients of our proof are the explicit blow-down function and a novel two-scale linearization in appropriate time-dependent spaces that yields uniform estimates in the perturbation parameter.