Some mathematical theories on the gas motion under the influence of external forcing

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 Navier-Stokes 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 Chapman-Enskog 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 high-order energy estimates with the spectral analysis. The same method is applied to the general time-dependent external force, especially the time-periodic one for which the existence and asymptotical stability of the time-periodic 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 Navier-Stokes 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 Navier-Stokes 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 Lp-Lq convergence rate of solutions to the Cauchy problem for the threedimensional Navier-Stokes equations with a potential force.