论文标题
与分数操作员的广义Cahn-Hilliard系统的渐近分析
An asymptotic analysis for a generalized Cahn-Hilliard system with fractional operators
论文作者
论文摘要
In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), 437-478 -- see also arXiv:1804.11290), the same authors have studied viscous and nonviscous Cahn-Hilliard systems of two operator equations in which nonlinearities of接受了双孔类型，例如常规或对数电势以及具有指示功能的非平滑电位。出现在系统方程式中的操作员是分数幂$ a^{2r} $和$ b^{2σ} $（从光谱中）（从光谱中）的一般线性运算符$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和$ b $，这些定义，无绑定，自selfadexhoint，selfdaist，以及在希尔伯特太空$ l^2（ω）中平稳$ω\ subset {\ mathbb {r}}^3 $，并具有紧凑的分解。引用的论文已经证明了存在，独特性和规律性结果。在这里，在粘性系统的情况下，我们将解决方案的渐近行为分析为运算符$ b^{2σ} $降低的参数$σ$趋向于零。我们证明在极限上融合了相位放松问题，我们还研究了这个限制问题，其中出现了一个额外的术语，其中包含$ b $的内核上相变的投影。
In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), 437-478 -- see also arXiv:1804.11290), the same authors have studied viscous and nonviscous Cahn-Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers $A^{2r}$ and $B^{2σ}$ (in the spectral sense) of general linear operators $A$ and $B$, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space $L^2(Ω)$, for some bounded and smooth domain $Ω\subset{\mathbb{R}}^3$, and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter $σ$ appearing in the operator $B^{2σ}$ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears.
