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We describe a time-dependent functional involving the relative entropy and the P H1 seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functional sheds light on the competition between dissipation and nonlinearity for this equation. It enables us to obtain new results concerning regularity/ blowup issues for the Landau equation with Coulomb potential. [ABSTRACT FROM AUTHOR]

In this paper, we extend to the case of initial data constituted of a Dirac mass plus a bounded density (with finite moments) the theory of Lions and Perthame [8] for the Vlasov–Poisson equation. Our techniques also provide polynomially growing in time estimates for moments of the bounded density. [ABSTRACT FROM AUTHOR]

Abstract: For the homogeneous Boltzmann equation with (cutoff or noncutoff) hard potentials, we prove estimates of propagation of norms with a weight (, large enough), as well as appearance of such weights. The proof is based on some new functional inequalities for the collision operator, proven by elementary means. [Copyright &y& Elsevier]