Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation
Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation
复制标题
Hele-Shaw 方程的 Lyapunov 函数、恒等式和柯西问题
DOI：
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复制
发表时间：
2019
影响因子：
2.4
通讯作者：
D. Smets
中科院分区：
文献类型：
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作者：
T. Alazard;N. Meunier;D. Smets
本文主要研究Hele-Shaw方程。我们介绍了一种方法的灵感来自水波理论。从减少到边界，引入狄利克雷诺依曼算子和利用各种取消，我们展示了抛物型发展方程的水平和垂直轨迹的自由表面上的速度。这允许以自然的方式准线性化方程。将这些精确恒等式与凸性不等式相结合，证明了不同性质的隐李雅普诺夫函数的存在性。我们还推导出从这些身份和以前的作品的水波问题的一个简单的证明的适定性的柯西问题。分析包含两个独立利益的侧面结果。首先，我们给出了一个原则，以获得估计的连续模的偏微分方程在一般假设下的流动。其次证明了Dirichlet to Neumann算子的一个凸性不等式并给出了应用。
This article is devoted to the study of the Hele–Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equations in a natural way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapunov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem. The analysis contains two side results of independent interest. Firstly, we give a principle to derive estimates for the modulus of continuity of a PDE under general assumptions on the flow. Secondly we prove and give applications of a convexity inequality for the Dirichlet to Neumann operator.
DOI：
10.1016/j.anihpc.2016.09.001
发表时间：
2017-07-01
影响因子：
1.9
作者：
Constantin, Peter;Gancedo, Francisco;Vicol, Vlad
通讯作者：
Vicol, Vlad
一些自由边界问题被改写为非局部抛物线方程
DOI：
10.1016/j.na.2019.05.019
发表时间：
2019
期刊：
Nonlinear Analysis
影响因子：
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作者：
Chang-Lara, Héctor A.;Guillen, Nestor;Schwab, Russell W.
通讯作者：
Schwab, Russell W.