论文标题
粘性Cahn-Hilliard系统的可溶解性和滑动模式控制可能具有奇异的潜力
Solvability and sliding mode control for the viscous Cahn-Hilliard system with a possibly singular potential
论文作者
论文摘要
在目前的贡献中，我们研究了一个粘性的cahn-hilliard系统，在该系统中，存在化学势$μ$的表达式中的进一步领先术语。该术语由$ l^2（ω）$中的一个细分运算符$ s $（其中$ω$是进化发生的域），该范围是在相位变量$φ$和给定状态$φ^* $的差的范围内，这是规定的，可能取决于空间和时间。在化学势$μ$的均质Neumann和Dirichlet边界条件下，我们证明存在和连续依赖性结果。接下来，假设$ s =ρ\，$ sign，$ sign运算符的倍数，并且对于更顺畅的数据，我们首先显示规律性结果。然后，在$μ$的Dirichlet边界条件下，在$ρ$和$ω$的适当条件下，我们还证明了滑动模式属性，也就是说，$φ$被迫在某个时候加入$φ^* $的演变$ t^* $，低于给定的最终时间$ t $。我们指出，我们所有的结果都适用于非常笼统的多孔潜力，该潜力在$φ$上。
In the present contribution we study a viscous Cahn-Hilliard system where a further leading term in the expression for the chemical potential $ μ$ is present. This term consists of a subdifferential operator $S$ in $L^2(Ω)$ (where $Ω$ is the domain where the evolution takes place) acting on the difference of the phase variable $φ$ and a given state $φ^* $, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential $μ$. Next, by assuming that $S=ρ\,$sign, a multiple of the sign operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for $μ$ and under suitable conditions on $ρ$ and $Ω$, we also prove the sliding mode property, that is, that $φ$ is forced to join the evolution of $φ^* $ in some time $T^*$ lower than the given final time $T$. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on $φ$.
