In this paper, we investigate the Cauchy problem for the compressible nematic liquid crystal flows in three-dimensional whole space. First of all, we establish the time decay rates for compressible nematic liquid crystal flows by the method of spectral analysis and energy estimates. Furthermore, we enhance the convergence rates for the higher-order spatial derivatives of density, velocity and director. Finally, the time decay rates of mixed space-time derivatives of solution are also established.
In this paper, we consider the well-posedness of the compressible nematic liquid crystal flow with the cylinder symmetry in R 3. By establishing a uniform pointwise positive lower and upper bounds of the density, we derive the global existence and uniqueness of strong solution and show the long time behavior of the global solution. Our results do not need the smallness of the initial data. Furthermore, a regularity result of global strong solution is given as well.
In this paper, we investigate the density-dependent incompressible nematic liquid crystal flows in n(n = 2 or 3) dimensional bounded domain. More precisely, we obtain the local existence and uniqueness of the solutions when the viscosity coefficient of fluid depends on density. Moreover, we establish blowup criterions for the regularity of the strong solutions in dimension two and three respectively. In particular, we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration. For these results, the initial density needs not be strictly positive.
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