Global well‐posedness for the one‐phase Muskat problem
Global well‐posedness for the one‐phase Muskat problem
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DOI：
10.1002/cpa.22124
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发表时间：
2021-03
影响因子：
3
通讯作者：
Hongjie Dong;F. Gancedo;Huy Q. Nguyen
中科院分区：
文献类型：
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作者：
Hongjie Dong;F. Gancedo;Huy Q. Nguyen
研究了渗流介质中二维流体的自由边界问题。这被称为单相马斯卡特问题，在数学上等价于重力驱动的垂直Hele-Shaw问题。证明了如果初始自由边界是周期Lipschitz函数的图形，则在强∞Lx2$L^\inty_t L^2_x$意义下存在全局-时间Lipschitz解，且它是唯一的粘性解.证明需要对Lipschitz域中的层势和逐点椭圆正则性进行定量估计。这是首次构造具有任意大小初始数据的Muskat问题的唯一全局强解。
The free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong Lt∞Lx2$L^\infty _t L^2_x$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.