TY - JOUR
T1 - Pointwise Description for the Linearized Fokker–Planck–Boltzmann Model
AU - Wu, Kung Chien
N1 - Funding Information:
The author would like to thank Professor Dr. Clément Mouhot for his encouragement and fruitful discussions concerning this paper when the author was a postdoctoral fellow in Cambridge. This work was supported by the Ministry of Science and Technology under the Grant 102-2115-M-017-004-MY2 and NCTS. Part of this work was written during the stay at Institute of Mathematics, Academia Sinica and Department of Mathematics, Stanford University; the author thanks Tai-Ping Liu for his kind hospitality.
Publisher Copyright:
© 2015, Springer Science+Business Media New York.
PY - 2015/9/1
Y1 - 2015/9/1
N2 - In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker–Planck–Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543–1608, 2004) for Boltzmann equation, but the Fokker–Planck term in this paper creates some technical difficulties.
AB - In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker–Planck–Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543–1608, 2004) for Boltzmann equation, but the Fokker–Planck term in this paper creates some technical difficulties.
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U2 - 10.1007/s10955-015-1206-0
DO - 10.1007/s10955-015-1206-0
M3 - Article
AN - SCOPUS:84938415768
VL - 160
SP - 1277
EP - 1293
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
SN - 0022-4715
IS - 5
ER -