TY - JOUR
T1 - Stability of the rarefaction wave of the vlasov-poisson-boltzmann system
AU - Duan, Renjun
AU - Liu, Shuangqian
N1 - Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to lbscholars@cityu.edu.hk.
PY - 2015
Y1 - 2015
N2 - This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov-Poisson-Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allow that the electric potential may take distinct constant states at both far fields. The rarefaction wave is constructed by the quasineutral Euler equations through the zero-order fluid dynamic approximation, and the wave strength is not necessarily small. We prove that the local Maxwellian with macroscopic quantities determined by the quasineutral rarefaction wave is time-asymptotically stable under small perturbations for the corresponding Cauchy problem. The main analytical tool is the combination of techniques we developed in [R.-J. Duan and S.-Q. Liu, J. Differential Equations, 258 (2015), pp. 2495-2530] for the viscous compressible fluid with the self-consistent electric field and the refined energy method based on the macro-micro decomposition of the Boltzmann equation around a local Maxwellian. Both the time decay property of the rarefaction waves and the structure of the system play a key role in the proof.
AB - This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov-Poisson-Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allow that the electric potential may take distinct constant states at both far fields. The rarefaction wave is constructed by the quasineutral Euler equations through the zero-order fluid dynamic approximation, and the wave strength is not necessarily small. We prove that the local Maxwellian with macroscopic quantities determined by the quasineutral rarefaction wave is time-asymptotically stable under small perturbations for the corresponding Cauchy problem. The main analytical tool is the combination of techniques we developed in [R.-J. Duan and S.-Q. Liu, J. Differential Equations, 258 (2015), pp. 2495-2530] for the viscous compressible fluid with the self-consistent electric field and the refined energy method based on the macro-micro decomposition of the Boltzmann equation around a local Maxwellian. Both the time decay property of the rarefaction waves and the structure of the system play a key role in the proof.
KW - Energy method
KW - Nonlinear stability
KW - Quasineutral rarefaction waves
KW - Vlasov-Poisson-Boltzmann system
UR - http://www.scopus.com/inward/record.url?scp=84947427332&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84947427332&origin=recordpage
U2 - 10.1137/140995179
DO - 10.1137/140995179
M3 - RGC 21 - Publication in refereed journal
VL - 47
SP - 3585
EP - 3647
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
SN - 0036-1410
IS - 5
ER -