Departamento de Análisis Matemático, Universidad de Sevilla, 41012 Sevilla, Spain.
Proc Natl Acad Sci U S A. 2014 Jan 14;111(2):635-9. doi: 10.1073/pnas.1320554111. Epub 2013 Dec 17.
In this paper, for both the sharp front surface quasi-geostrophic equation and the Muskat problem, we rule out the "splash singularity" blow-up scenario; in other words, we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem. Our result confirms the numerical simulations in earlier work, in which it was shown that the curvature blows up because the contours collapse at a point. Here, we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out; in this case, a positive volume of fluid between the contours cannot be ejected in finite time.
在本文中，针对锐前线准地转方程和 Muskat 问题，我们排除了“喷溅奇点”爆炸场景；换句话说，我们证明了无论从哪个系统演化的轮廓线在自由边界保持光滑时都不能在单点相交。喷溅奇点在自由边界不可压缩 Euler 方程的形式，也就是水波轮廓演化问题中成立。我们的结果证实了早期工作中的数值模拟，其中表明曲率的爆炸是因为轮廓在一个点上崩溃。在这里，我们证明了控制曲率将消除相间崩溃的可能性。我们提供的另一个结论是对早期排除射流奇点的工作的更好理解；在这种情况下，两个轮廓之间的正体积的流体不能在有限的时间内排出。
