TY - JOUR
T1 - Uncertainty principle and kinetic equations
AU - Alexandre, R.
AU - Morimoto, Y.
AU - Ukai, S.
AU - Xu, C. J.
AU - Yang, T.
PY - 2008/10/15
Y1 - 2008/10/15
N2 - A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation. © 2008 Elsevier Inc. All rights reserved.
AB - A large number of mathematical studies on the Boltzmann equation are based on the Grad's angular cutoff assumption. However, for particle interaction with inverse power law potentials, the associated cross-sections have a non-integrable singularity corresponding to the grazing collisions. Smoothing properties of solutions are then expected. On the other hand, the uncertainty principle, established by Heisenberg in 1927, has been developed so far in various situations, and it has been applied to the study of the existence and smoothness of solutions to partial differential equations. This paper is the first one to apply this celebrated principle to the study of the singularity in the cross-sections for kinetic equations. Precisely, we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations. In particular, we give some estimates on the regularity of solutions in Sobolev spaces w.r.t. all variables for both linearized and nonlinear space inhomogeneous Boltzmann equations without angular cutoff, as well as the linearized space inhomogeneous Landau equation. © 2008 Elsevier Inc. All rights reserved.
KW - Boltzmann equations
KW - Kinetic equations
KW - Landau equation
KW - Microlocal analysis
KW - Non-cutoff cross-sections
KW - Uncertainty principle
UR - http://www.scopus.com/inward/record.url?scp=56349126806&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-56349126806&origin=recordpage
U2 - 10.1016/j.jfa.2008.07.004
DO - 10.1016/j.jfa.2008.07.004
M3 - RGC 21 - Publication in refereed journal
VL - 255
SP - 2013
EP - 2066
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 8
ER -