Mathematik in den Naturwissenschaften Leipzig On a Diffuse Interface Model for Two-Phase Flows of Viscous , Incompressible Fluids with Matched Densities
Mathematik in den Naturwissenschaften Leipzig On a Diffuse Interface Model for Two-Phase Flows of Viscous , Incompressible Fluids with Matched Densities
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发表时间：
2007
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通讯作者：
H. Abels
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作者：
H. Abels
研究了两种相同密度的粘性不可压缩牛顿流体在有界区域内流动的扩散界面模型。流体被假定为宏观上是不混溶的，但在一个小的界面区域的部分混合被假定在模型中。此外，考虑到这两种成分的扩散。这导致了耦合的Navier-Stokes/Cahn-Hilliard系统，它能够描述液滴的形成和碰撞过程中的流动的演变。我们证明了存在的弱解的非平稳系统在两个和三个空间维度的一类物理相关的和奇异的自由能密度，这确保了-在通常情况下，一个光滑的自由能密度-的浓度停留在物理合理的区间。此外，我们给出了弱解的正则性和唯一性的一些结果。特别是，我们得到的唯一的“强”的解决方案存在于二维全球的时间和三维的时间局部。此外，我们还证明了对于任何弱解，浓度在空间和时间上是一致连续的。由于这种正则性，我们能够证明任何弱解在很长时间内都是正则的，并且当t → ∞时收敛到平稳系统的解。这些结果是基于分数时间正则性空间中具有对流和奇异势的Cahn-Hilliard方程的正则性理论以及L2（0，∞; H（Ω）），s ∈ [0，1]中具有变粘性和力的Stokes系统的最大正则性的一个新结果。
We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids of the same density in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. This leads to a coupled Navier-Stokes/Cahn-Hilliard system, which is capable to describe the evolution of droplet formation and collision during the flow. We prove existence of weak solutions of the nonstationary system in two and three space dimensions for a class of physical relevant and singular free energy densities, which ensures – in contrast to the usual case of a smooth free energy density – that the concentration stays in the physical reasonable interval. Furthermore, we present some results on regularity and uniqueness of weak solutions. In particular, we obtain that unique “strong” solutions exist in two dimensions globally in time and in three dimensions locally in time. Moreover, we show that for any weak solution the concentration is uniformly continuous in space and time. Because of this regularity, we are able to show that any weak solution becomes regular for large times and converges as t → ∞ to a solution of the stationary system. These results are based on a regularity theory for the Cahn-Hilliard equation with convection and singular potentials in spaces of fractional time regularity as well as on a result on maximal regularity of a Stokes system with variable viscosity and forces in L2(0,∞; H(Ω)), s ∈ [0, 1], which are new themselves.