具有中等斜率的三维Muskat问题的整体适定性
Global wellposedness for the 3D Muskat problem with medium size slope
摘要：
证明了当初始界面具有次线性增长和斜率$||\nabla_x f_0||_{L^\infty}< 5^{-1/2}$时，三维Muskat问题在稳定区域的整体经典解的存在唯一性.在这些假设下，我们证明了方程基本上是抛物型的，满足比较原理。应用连续模技术，我们表明，粗糙的初始数据立即成为$C^{1,1}$与曲率衰减一样$O(t^{-1})$。
英文摘要：
We prove the existence and uniqueness of global, classical solutions to the 3D Muskat problem in the stable regime whenever the initial interface has sublinear growth and slope $||\nabla_x f_0||_{L^\infty}< 5^{-1/2}$. We show under these assumptions that the equation is fundamentally parabolic, satisfying a comparison principle. Applying the modulus of continuity technique, we show that rough initial data instantly becomes $C^{1,1}$ with the curvature decaying like $O(t^{-1})$.
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