We prove a Schauder estimate for kinetic Fokker-Planck equations that requires only Hölder regularity in space and velocity but not in time. As an application, we deduce a weakstrong uniqueness result of classical solutions to the spatially inhomogeneous Landau equation beginning from initial data having Hölder regularity in x and only a logarithmic modulus of continuity in v. This replaces an earlier result requiring Hölder continuity in both variables.
For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in the whole space R 3x , we establish pointwise lower bounds that appear instantaneously even if the initial data contains vacuum regions. Our lower bounds depend only on the initial data and upper bounds for the mass and energy densities of the solution. As an application, we improve the weakest known continuation criterion for large-data solutions, by removing the assumptions of mass bounded below and entropy bounded above.where b is uniformly positive and bounded. * :At this point, we may quote verbatim the calculations of [11, Lemma 3.4] to obtain
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.