Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential (Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations)

Abstract

This report succinctly summarizes results proved in the authors' recent work [7] where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space A(Ω), which can be expected to play a similar role to L ∞. Compared to the L2-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.