Addendum to “A Consideration of the Morphological Stability of an Interface”

Abstract

In our previous paper, we discuss the flattening properties of a generalized interface through the harmony between phenomenal facts and a mathematical structure, and we propose an original principle of interface morphology: ‘‘The interface tends to become flat if the driving force decreases with the increase of the curvature’’. However, we see the incompleteness of its universality; that is, a question ‘‘Is the principle applied even if the contribution neglecting the curvature effect AI or AII or the curvature effect fI or fII depends not only on the height of the interface u but also on the time t or the spatial variable x1 or x2 caused by concentration distribution, impurity distribution, temperature distribution and so on?’’ naturally comes to mind. Then, letting AI, AII, fI, and fII be a function of t, x1, x2, and u, we verify the principle in the same way as that in our previous study. First, with the aid of the implicit function theorem and the mean value theorem, the notion of a weak flattening property is acquired as follows. 1) Even if AI, AII, fI, and fII actually depend on t, x, y, and u, the velocity of the peak top (the valley bottom) of the interface is lower (greater) than the function of t; x; y, and u, which corresponds to the velocity without the curvature effect. Then, with the aid of the lemmas in our previous paper on a nonlinear evolution equation and an ordinary differential equation, the notion of the hierarchy of flattening properties is acquired as follows. 2) If AI and AII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of t, x2 (or x1), and u, which corresponds to the velocity without the curvature effect. 3) If AI, AII, fI, and fII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of t, x2 (or x1), and u, which corresponds to the velocity without the curvature effect. Then, the bifurcation never takes place. 4) If AI and AII are independent of x1, x2 and u and fI and fII are independent of x1 (or x2), the velocity of the peak top (the valley bottom) is lower (greater) than the function of only t, which corresponds to the velocity of a plane interface. In particular, if AI and AII are constant, the function is constant. Then, the bifurcation never takes place. 5) If AI and AII depend only on u [that is, AIðt; x; y; uÞ 1⁄4 ~ AIðuÞ;AIIðt; x; y; uÞ 1⁄4 ~ AIIðuÞ] under the condition 9 s.t. ~ AIð Þ 1⁄4 ~ AIIð Þ, d1⁄2 ~ AIðuÞ ~ AIIðuÞ = duju1⁄4 < 0, which reinforces the flattening property, and fI and fII are independent of x1 (or x2), the interface exponentially approaches a flat plane u 1⁄4 without the bifurcation. Consequently, the difference between the role of the contributions neglecting the curvature effects (AI and AII) and that of the curvature effects ( fI and fII) for the flattening properties is revealed, and the above principle is more universally exhibited.