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This work deals with the large time behaviour of the spatially homogeneous Landau equation with Coulomb potential. Firstly, we obtain a bound from below of the entropy dissipation D( f) by weighted relative Fisher information of f with respect to the associated Maxwellian distribution, which leads to a variant of Cercignani's conjecture thanks to a logarithmic Sobolev inequality. Secondly, we prove the propagation of polynomial and stretched exponential moments with an at-most linearly growing in-time rate. As an application of these estimates, we show the convergence of any ( H- or weak) solution to the Landau equation with Coulomb potential to the associated Maxwellian equilibrium with an explicitly computable rate, assuming initial data with finite mass, energy, entropy and some higher L -moment. More precisely, if the initial data have some (large enough) polynomial L -moment, then we obtain an algebraic decay. If the initial data have a stretched exponential L -moment, then we recover a stretched exponential decay. [ABSTRACT FROM AUTHOR]

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering all physical collision kernels). These estimates are conditional on some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non-cutoff and non-mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions. [ABSTRACT FROM AUTHOR]

In this work, we consider the smoothness of the solutions to the full Landau equation. In particular, we prove that any classical solutions (such as the ones obtained by Guo in a “close to equilibrium” setting) become immediately smooth with respect to all variables. This shows that the Landau equation is a nonlinear and nonlocal analog of an hypoelliptic equation. [ABSTRACT FROM AUTHOR]