We consider the hyperbolic relaxation of the viscous Cahn–Hilliard equation

in a bounded domain of \({\mathbb R}^d\) with smooth boundary when subject to Neumann boundary conditions, and on rectangular domains when subject to periodic boundary conditions. The space dimension is \(d=1,\) 2 or 3, but it is required \(\delta =\varepsilon =0\) when \(d=2\) or 3; \(\delta \) being the viscosity parameter. The constant \(\varepsilon \in (0,1]\) is a relaxation parameter, \(\phi \) is the order parameter and \(g:{\mathbb R}\rightarrow {\mathbb R}\) is a nonlinear function. This equation models the early stages of spinodal decomposition in certain glasses. Assuming that \(\varepsilon \) is dominated from above by \(\delta \) when \(d=2\) or 3, we construct a family of exponential attractors for Eq. (0.1) which converges as \((\varepsilon ,\delta )\) goes to \((0,\delta _0),\) for any \(\delta _0\in [0,1],\) with respect to a metric that depends only on \(\varepsilon \), improving previous results where this metric also depends on \(\delta \). Then we introduce two change of variables and corresponding problems, in the case of rectangular domains and \(d=1\) or 2 only. First, we set \(\tilde{\phi }(t)=\phi (\sqrt{\varepsilon } t)\) and we rewrite Eq. (0.1) in the variables \((\tilde{\phi },\tilde{\phi }_t).\) We show that there exist an integer n, independent of both \(\varepsilon \) and \(\delta \), a value \(0<\tilde{\varepsilon }_0(n)\le 1\) and an inertial manifold of dimension n, for either \(\varepsilon \in (0,\tilde{\varepsilon }_0]\) and \(\delta =2\sqrt{\varepsilon }\) or \(\varepsilon \in (0,\tilde{\varepsilon }_0]\) and \(\delta \in [0,3\varepsilon ]\). Then, we prove the existence of an inertial manifold of dimension that depends on \(\varepsilon \), but is independent of \(\delta \) and \(\eta \), for any fixed \(\varepsilon \in (0,(\eta -2)^2]\) and every \(\delta \in [\varepsilon ,(2-\eta )\sqrt{\varepsilon }]\), for an arbitrary \(\eta \in (1,2)\). Next, we show the existence of an inertial manifold of dimension that depends on \(\varepsilon \) and \(\eta \), but is independent of \(\delta \), for any fixed \(\varepsilon \in (0,\frac{1}{(2+\eta )^2}]\) and every \(\delta \in [(2+\eta )\sqrt{\varepsilon },1]\), where \(\eta >0\) is arbitrary chosen. Moreover, we show the continuity of the inertial manifolds at \(\delta =\delta _0,\) for any \(\delta _0\in [0,(2-\eta )\sqrt{\varepsilon }]\cup [(2+\eta )\sqrt{\varepsilon },1]\). Second, we set \(\phi _t=-(2\varepsilon )^{-1}(I-\delta \Delta )\phi +\varepsilon ^{-1/2}v\) and we rewrite Eq. (0.1) in the variables \((\phi ,v)\). Then, we prove the existence of an inertial manifold of dimension that depends on \(\delta \), but is independent of \(\varepsilon ,\) for any fixed \(\delta \in (0,1]\) and every \(\varepsilon \in (0,\frac{3}{16}\delta ^2]\). In addition, we prove the convergence of the inertial manifolds when \(\varepsilon \rightarrow 0^+\).

我们考虑粘性Cahn-Hilliard方程的双曲松弛

在受Neumann边界条件约束时，在具有平滑边界的\（{{mathbb R} ^ d \）的有界域中；在受周期边界条件约束时，在矩形域中。空间尺寸为\（d = 1，\） 2或3，但是当\（d = 2 \）或3时，空间尺寸为\（\ delta = \ varepsilon = 0 \）；\（\ delta \）是粘度参数。常量\（\ varepsilon \ in（0,1] \）是松弛参数，\（\ phi \）是阶数参数，\（g：{\ mathbb R} \ rightarrow {\ mathbb R} \）是非线性函数，该方程对某些眼镜中旋节线分解的早期阶段进行建模，假设\（\ varepsilon \）当\（d = 2 \）或3时，从上方由\（\ delta \）主导，我们为Eq构造了一个指数吸引子族。（0.1），其收敛为\（（\ varepsilon，\三角洲）\）进入\（（0，\增量_0），\）对于任何\（\在[0,1]增量_0 \，\）相对于到仅依赖于\（\ varepsilon \）的度量，改进了以前的结果，该度量也依赖于\（\ delta \）。然后，在矩形域和\（d = 1 \）或仅2的情况下，我们介绍了变量的两个变化和相应的问题。首先，我们设置\（\ tilde {\ phi}（t）= \ phi（\ sqrt {\ varepsilon} t）\），然后重写Eq。（0.1）中的变量\（（（\ tilde {\ phi}，\ tilde {\ phi} _t）。\）我们表明存在一个整数n，它独立于\（\ varepsilon \）和\（\ delta \）（一个值\ （0 <\ tilde {\ varepsilon} _0（n）\ le 1 \）和大小为n的惯性流形，对于\（\ varepsilon \ in（0，\ tilde {\ varepsilon} _0] \）和\（ \ delta = 2 \ sqrt {\ varepsilon} \）或\（\ varepsilon \ in（0，\ tilde {\ varepsilon} _0] \）和\（\ delta \ in [0,3 \ varepsilon] \）。 ，我们证明存在一个惯性流形，其大小取决于\（\ varepsilon \），但与\（\ delta \）和\（\ eta \），对于任何固定的\（\ varepsilon \ in（0，（\ eta -2）^ 2] \）和每个\（\ delta \ in [\ varepsilon，（2- \ eta）\ sqrt {\ varepsilon}] \），对于任意\（\ eta \ in（1,2）\）。接下来，我们说明存在依赖于\（\ varepsilon \）和\（\ ETA \） ，但是独立\（\三角洲\） ，对于任何固定的\（\ varepsilon \在（0，\压裂{1} {（2 + \ ETA）^ 2}] \）和每\（\ delta \ in [（2+ \ eta）\ sqrt {\ varepsilon}，1] \），其中\（\ eta> 0 \）是任意选择的，此外，我们显示了惯性流形在\（\ delta处的连续性= \增量_0，\）对于任何\（\ delta _0 \ in [0，（2- \ eta）\ sqrt {\ varepsilon}] \ cup [（2+ \ eta）\ sqrt {\ varepsilon}，1] \）。其次，我们设置\（\ phi _t =-（2 \ varepsilon）^ {-1}（I- \ delta \ Delta）\ phi + \ varepsilon ^ {-1/2} v \）并重写Eq。（0.1）在变量\（（\ phi，v）\）中。然后，我们证明尺寸的惯性歧管，其取决于存在\（\三角洲\） ，但是独立\（\ varepsilon，\）对于任何固定在（0,1] \ \（\三角洲\）和每个\（\ varepsilon \ in（0，\ frac {3} {16} \ delta ^ 2] \）此外，我们证明了当\（\ varepsilon \ rightarrow 0 ^ + \）时惯性流形的收敛性。