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ABSTRACT We investigate a kinetic model for the sedimentation of dilute suspensions of rod-like particles under gravity, deduced by Helzel, Otto, and Tzavaras (2011), which couples the impressible (Navier-)Stokes equation with the Fokker-Planck equation. With a no-flux boundary condition for the distribution function, we establish the existence and uniqueness of a global weak solution to the two dimensional model involving the Stokes equation.

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In this paper we establish the local-in-time existence and uniqueness of strong solutions to the free boundary problem of the full compressible Navier-Stokes equations in three-dimensional space. The vanishing density and temperature condition is imposed on the free boundary, which captures the motions of the non-isentropic viscous gas surrounded by vacuum with bounded entropy. We also assume some proper decay rates of the density towards the boundary and singularities of derivatives of the temperature across the boundary on the initial data, which coincides with the physical vacuum condition for the isentropic flows. This extends the previous result of Liu [arXiv: 1612.07936] by removing the spherically symmetric assumption and considering more general initial density and temperature profiles.

There are many open problems on the stability of nonlinear wave patterns to the Boltzmann equation even though the corresponding stability theory has been comparatively well-established for the gas dynamical systems. In this paper, we study the nonlinear stability of a rarefaction wave profile to the Boltzmann equation with the boundary effect imposed by specular reflection for both the hard sphere model and the hard potential model with angular cut-off. The analysis is based on the property of the solution and its derivatives which are either odd or even functions at the boundary coming from specular reflection, and the decomposition on both the solution and the Boltzmann equation introduced in [24, 26] for energy method.