TY - JOUR
T1 - A combination of energy method and spectral analysis for study of equations of gas motion
AU - Duan, Renjun
AU - Ukai, Seiji
AU - Yang, Tong
PY - 2009/6
Y1 - 2009/6
N2 - There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system. © 2009 Higher Education Press and Springer-Verlag GmbH.
AB - There have been extensive studies on the large time behavior of solutions to systems on gas motions, such as the Navier-Stokes equations and the Boltzmann equation. Recently, an approach is introduced by combining the energy method and the spectral analysis to the study of the optimal rates of convergence to the asymptotic profiles. In this paper, we will first illustrate this method by using some simple model and then we will present some recent results on the Navier-Stokes equations and the Boltzmann equation. Precisely, we prove the stability of the non-trivial steady state for the Navier-Stokes equations with potential forces and also obtain the optimal rate of convergence of solutions toward the steady state. The same issue was also studied for the Boltzmann equation in the presence of the general time-space dependent forces. It is expected that this approach can also be applied to other dissipative systems in fluid dynamics and kinetic models such as the model system of radiating gas and the Vlasov-Poisson-Boltzmann system. © 2009 Higher Education Press and Springer-Verlag GmbH.
KW - Asymptotic stability
KW - Energy method
KW - Linearization
KW - Optimal rate
KW - Spectral analysis
UR - http://www.scopus.com/inward/record.url?scp=63849275800&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-63849275800&origin=recordpage
U2 - 10.1007/s11464-009-0020-x
DO - 10.1007/s11464-009-0020-x
M3 - RGC 21 - Publication in refereed journal
SN - 1673-3452
VL - 4
SP - 253
EP - 282
JO - Frontiers of Mathematics in China
JF - Frontiers of Mathematics in China
IS - 2
ER -