Some mathematical theories on the gas motion under the influence of external forcing
一些關於外力影響下氣體运動的數學理論
Student thesis: Doctoral Thesis
Author(s)
Related Research Unit(s)
Detail(s)
Awarding Institution  

Supervisors/Advisors 

Award date  2 Oct 2008 
Link(s)
Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(58f531a3033e438caf4925ab90a251cf).html 

Other link(s)  Links 
Abstract
This thesis is concerned with the mathematical study of the gas motion under the influence
of external forcing. The models considered are the Boltzmann equation in the
kinetic theory and the compressible NavierStokes equations in the fluid dynamics,
which have a close relation in the sense that the latter can be derived as an approximation
of second order from the former through the ChapmanEnskog expansion.
In the first part, the Cauchy problems on the Boltzmann equation near vacuum or
Maxwellians are investigated for the case when the external forces are present. Global
existence and uniform in time stability of solutions are proved in the framework of
small perturbations. Moreover, the optimal rate of convergence of the solution to the
Maxwellian is obtained by combining the refined highorder energy estimates with the
spectral analysis. The same method is applied to the general timedependent external
force, especially the timeperiodic one for which the existence and asymptotical stability
of the timeperiodic solution with the same period is proved if the spatial dimension
is not less than five.
In the second part, two mathematical results about the compressible NavierStokes
equations with external forces are obtained. One is the global existence and uniqueness
of weak solutions to the initial boundary value problem for the one dimensional
isentropic NavierStokes equations under the gravitational force when the viscosity
depends on the density and the initial density is continuously connected to vacuum.
The other one, whose proof is similar as in the case of the Boltzmann equation, is
the optimal LpLq convergence rate of solutions to the Cauchy problem for the threedimensional
NavierStokes equations with a potential force.
 Fluid dynamics, Mathematical models, Kinetic theory of gases