The Cahn-Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity
The Cahn-Hilliard Equation with Forward-Backward Dynamic Boundary Condition via Vanishing Viscosity
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DOI：
10.1137/21m142441x
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发表时间：
2021-06
期刊：
影响因子：
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通讯作者：
P. Colli;T. Fukao;Luca Scarpa
中科院分区：
文献类型：
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作者：
P. Colli;T. Fukao;Luca Scarpa
对一个具有Cahn-Hilliard型动力边界条件的方程组，当作用于相变量的表面扩散系数趋于0时，进行了渐近分析，得到了一个极限的正向后动力边界条件.这是在一个非常一般的设置，与非线性项承认最大单调图在批量和边界上。这两个图通过一个增长条件相关联，其中边界图支配另一个。结果表明，在极限过程中，问题的解失去了一些规律性，极限方程必须在次微分包含的意义下得到适当的解释。然而，极限问题仍然是适定的，因为可以证明一个连续的依赖估计。此外，当两个图具有相同的增长性时，其解具有更强的正则性，边界条件几乎处处成立。对于扩散参数的适当阶数，还可以示出误差估计。
An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn–Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition, with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem looses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown, for a suitable order of the diffusion parameter.