Uncertainty principle and kinetic equations
The main aim of the paper is to apply uncertainty principle to the study of smoothing effects arising from the non-cutoff cross-sections for the space inhomogeneous kinematic equations. Generalized version of the uncertainty principle is proved. Based on this version results on the smoothing effect on solutions to the kinematic equations are proved, by analyzing the interaction between the transport operator and the regularity assumption in the microscopic velocity, together with a mild regularity assumption on the source term. First the regularity for transport equation is regarded, then existence and regularity of solutions for linearized Boltzmann equation and linearized Landau equation are proved. Moreover, the regularity result for nonlinear Boltzmann equation is also proved.
scientific article; zbMATH DE number 5480725
scientific article