论文标题
凸性和Hele-Shaw方程
Convexity and the Hele-Shaw equation
论文作者
论文摘要
沃尔特·克雷格（Walter Craig）在水波问题上的开创性作品确立了几种确切身份的重要性：扎卡罗夫的汉密尔顿公式，对迪里奇特到尼曼操作员的形状导数公式，正常形式的变换。在本文中，我们为Hele-Shaw方程式介绍了几种身份，这些身份受他的非线性方法的启发。首先，我们研究未知数的凸变化，并获得大量的强Lyapounov函数。除了不侵害外，这些lyapounov函数是时间的凸函数。该分析依赖于Hele-Shaw方程的简单椭圆形公式，该公式具有独立感兴趣。然后，我们研究了凸性控制溶液空间衍生物的作用。我们考虑雷利 - 泰勒系数$ a $的演化方程（这是与自由表面压力的正常衍生物成正比的正函数）。受到椭圆形或抛物线方程熵的研究的启发，我们考虑了特殊功能$φ（x）= x \ log x $，并发现$φ（1/\ sqrt {a}）$是一个适合的方程式的子解决方案。作为一个应用程序，我们提供了最大principe的另一个证明，价格为$ 1/a $，$ 1/a $与尼古拉斯·梅尼尔（Nicolas Meunier）和迪迪埃（Didier）Smets获得。
Walter Craig's seminal works on the water-waves problem established the importance of several exact identities: Zakharov's hamiltonian formulation, shape derivative formula for the Dirichlet-to-Neumann operator, normal forms transformations. In this paper, we introduce several identities for the Hele-Shaw equation which are inspired by his nonlinear approach. Firstly, we study convex changes of unknowns and obtain a large class of strong Lyapounov functions; in addition to be non-increasing, these Lyapounov functions are convex functions of time. The analysis relies on a simple elliptic formulation of the Hele-Shaw equation, which is of independent interest. Then we study the role of convexity to control the spatial derivatives of the solutions. We consider the evolution equation for the Rayleigh-Taylor coefficient $a$ (this is a positive function proportional to the opposite of the normal derivative of the pressure at the free surface). Inspired by the study of entropies for elliptic or parabolic equations, we consider the special function $φ(x)=x\log x$ and find that $φ(1/\sqrt{a})$ is a sub-solution of a well-posed equation. As an application, we give another proof of the maximum principe for $1/a$ obtained with Nicolas Meunier and Didier Smets.
