This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with small angular velocity. In particular, the initial perturbation which preserves the angular momentum would decay exponentially in time and the solution to the Navier-Stokes equations converges to the steady state as time grows up.

In this paper, we study the Li stability of perturbation of constant states for 2 x 2 systems of conservation laws. We introduce a nonlinear functional which is equivalent to the Li norm of the difference between the constant state and the approximate solutions consisting of elementary waves and is non-increasing in time for the limiting weak solution. This yields the Li stability of the constant states. The approximate solutions are constructed with the aid of wave tracing for the random choice method. Our functional reveals the aspects of nonlinear wave behavior, particularly the coupling of waves pertaining to different characteristic families, which affects the Li norm of the solutions.

Although the decay in time estimates of the semi-group generated by the linearized Boltzmann operator without forcing have been well established, there is no corresponding result for the case with general external force. This paper is mainly concerned with the optimal decay estimates on the solution operator in some weighted Sobolev spaces for the linearized Boltzmann equation with a time dependent external force. No time decay assumption is made on the force. The proof is based on both the energy method through the macro-micro decomposition and the L p-L q estimates from the spectral analysis. The decay estimates thus obtained are applied to the study on the global existence of the Cauchy problem to the nonlinear Boltzmann equation with time dependent external force and source. Precisely, for space dimension n>= 3, the global existence and decay rates of solutions to the Cauchy problem are

We study the convergence rate of Glimm scheme for general systems of hyperbolic conservation laws without the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate. We first give a sharper estimate of the error arising from the wave tracing argument by a careful analysis of the interaction between small waves. With this key estimate, the convergence rate is shown to be o (1) s 1 3| ln s| 1+ , which is sharper compared to o (1) s 1 4| ln s| given in [T. Yang, Convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, Taiwanese J. Math. 7 (2)(2003) 195205]. However, it is still slower than o (1) s 1 2| ln s| given in [A. Bressan, A. Marson, Error bounds for a deterministic version of the Glimm scheme, Arch. Ration. Mech. Anal. 142 (2)(1998) 155176] for systems with each characteristic field being genuinely nonlinear or linearly degenerate. Here s is the

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