In this paper, we consider one-dimensional compressible viscous and heat-conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in .H 1 .0, 1// 4 , and then we establish the global existence and exponential stability of solutions in .H 2 .0, 1// 4 under the suitable assumptions for initial data. The results in this paper improve those previously related results.
In this paper, we consider one-dimensional compressible isentropic Navier-Stokes equations with the viscosity depending on density and with free boundary. The viscosity coefficient μ is proportional to ρ θ with 0 < θ < 1, where ρ is the density. The existence and uniqueness of global weak solutions in H 1 ([0, 1]) have been established in [S. Jiang, Z. Xin, P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal. 12 (2005) 239-252]. We will establish the regularity of global solution under certain assumptions imposed on the initial data by deriving some new a priori estimates.
In this paper, we prove the regularity and exponential stability of solutions in Hi (i = 2, 4) for a pth power Newtonian fluid undergoing one-dimensional longitudinal motions. Some new ideas and more delicate estimates are introduced to prove these results.
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