The Cahn–Hilliard–Oono equation with singular potential
The Cahn–Hilliard–Oono equation with singular potential
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DOI：
10.1142/s0218202517500506
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发表时间：
2017-10
影响因子：
3.5
通讯作者：
A. Giorgini;M. Grasselli;A. Miranville
中科院分区：
文献类型：
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作者：
A. Giorgini;M. Grasselli;A. Miranville
我们在ℝd，d≤3的有界域上考虑了具有奇异(如对数)势的所谓的Cahn-Hilliard-Oono方程。该方程受初始条件和Neumann齐次边界条件的约束。然而，与卡恩-希利亚德方程相反，总质量可能不守恒。Miranville和Temam证明了这类问题整体有限能量解的存在性。我们首先建立了该(唯一)解在有限时间内的正则化性质。然后，在第二维，我们证明了所谓的严格分离性质，即我们证明了任何有限能量解相对于初始能量和总质量一致地远离纯相。利用这些结果，我们研究了解的长期行为。更准确地说，我们在二维和三维两个维度上证明了全局吸引子的存在性。由于在二维空间中具有严格的分离性质。
We consider the so-called Cahn–Hilliard–Oono equation with singular (e.g. logarithmic) potential in a bounded domain of ℝd, d ≤ 3. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn–Hilliard equation, the total mass might not be conserved. The existence of a global finite energy solution to such a problem was proven by Miranville and Temam. We first establish some regularization properties in finite time of the (unique) solution. Then, in dimension two, we prove the so-called strict separation property, namely, we show that any finite energy solution stays away from pure phases, uniformly with respect to the initial energy and the total mass. Taking advantage of these results, we study the long-time behavior of solutions. More precisely, we establish the existence of the global attractor in both two and three dimensions. Due to the strict separation property in dimension tw...