English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the spatially inhomogeneous Landau equation with soft potentials,including the case of Coulomb interactions. First, we establish the existenceof solutions for a short time, assuming the initial data is in a fourth-orderSobolev space and has Gaussian decay in the velocity variable (no decayassumptions are made in the spatial variable). Next, we show that the evolutioninstantaneously spreads mass to every point in its domain. The resultingpointwise lower bounds have a sub-Gaussian rate of decay, which we show isoptimal. The proof of mass-spreading is based on a stochastic processassociated to the equation, and makes essential use of nonlocality. Bycombining this theorem with prior regularity results, we derive two importantapplications: $C^\infty$ smoothing in all three variables, even for initialdata with vacuum regions, and a continuation criterion that states the solutioncan be extended for as long as the mass and energy densities stay bounded fromabove. This is the weakest condition known to prevent blow-up. In particular,it does not require a lower bound on the mass density or an upper bound on theentropy density.

Local existence, lower mass bounds, and a new continuation criterion for the Landau equation