In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), 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 \(B^{2\sigma }\) (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space \(L^2(\Omega )\), for some bounded and smooth domain \(\Omega \subset {{\mathbb {R}}}^3\), 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 \(\sigma \) appearing in the operator \(B^{2\sigma }\) 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.

在最近的论文“广义分数Cahn-Hilliard系统的适定性和正则性”（Colli等人在Atti Accad Naz Lincei Rend Lincei Mat Appl 30：437-478，2019）中，同一作者研究了粘性和非粘性具有两个算子方程式的Cahn–Hilliard系统，其中承认了双阱类型的非线性（如正则或对数电势）以及具有指标函数的非光滑电势。在系统方程式中出现的算子是一般线性算子A和 B的分数幂\（A ^ {2r} \）和\（B ^ {2 \ sigma} \）（在频谱意义上），它们是密集定义的， Hilbert空间\（L ^ 2（\ Omega）\）中的无界，自伴和单调，适用于某些有界且平滑的域\（\ Omega \ subset {{\ mathbb {R}}} ^ 3 \），并且具有紧凑的解析器。引用的论文证明了存在性，唯一性和规律性结果。在这里，在粘性系统的情况下，我们分析解的渐近行为，因为出现在算子\（B ^ {2 \ sigma} \）中的参数\（\ sigma \）趋于零。我们证明了在极限处收敛到相位松弛问题，并且我们还研究了这个极限问题，其中出现了一个附加项，其中包含B内核上的相位变量的投影。