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In this paper, we review some recent results on the Boltzmann equation near the equilibrium states in the whole space $\mathbb R^n$. The emphasize is put on the well-posedness of the solution in some Sobolev space without time derivatives and its uniform stability and optimal decay rates, and also on the existence and asymptotical stability of the time-periodic solution. Most of results obtained here are proved by combining the energy estimates and the spectral analysis.
We present an $L^1$-asymptotic completeness results for relativistic kinetic equations with short range interaction forces. We show that the uniform phase space-time bound for nonlinear terms to the relativistic nonlinear kinetic equations yields the asymptotic completeness of the relativistic kinetic equations. For this space-time bound, we employ dispersive estimates and explicit construction of a Lyapunov functional.
In this paper, we study deformation of surfaces induced by adding one and two extra space variables to the motions of space curves in higher-dimensional similarity geometries. It is shown that the 2+1- and 3+1-dimensional nonlinear evolution equations including the 2+1-dimensional mKdV equation and a generalization to the mKdV-Burgers system arise from such motions.
Using method of matched asymptotic expansions, we derive the sharp interface limit for the diffusive interface model with the generalized Navier boundary condition recently proposed by Qian, Wang and Sheng in "Molecular scale contact line hydrodynamics of immiscible flows," and "Power-law slip profile of the moving contact line in two-phase immiscible flows," for the moving contact line problem. We show that the leading order problem satisfies a boundary value problem for a coupled Hale-Shaw and Navier-Stokes equations with the interface being a free boundary, and the leading order dynamic contact angle is the same as the static one satisfying the Young’s equation.