论文标题
Muskat问题的消失表面张力极限
The vanishing surface tension limit of the Muskat problem
论文作者
论文摘要
在一般环境中，Muskat问题涉及多孔培养基中不同密度和粘度的两种不可压缩流体之间的界面演变。界面运动是由重力和毛细管力驱动的，后者是由于表面张力引起的。为了领先的顺序，在Sobolev空间中，有和没有表面张力效应的Muskat问题都在扩展不变。我们证明，对于满足雷利 - 泰勒条件的任何亚临界数据，马斯卡特问题的解决方案与表面张力$ \ frak {s} $收敛到Muskat问题的独特解决方案，而没有及时与速率$ \ sqrt {\ frak {\ frak {s} $ n时$ \ frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak frak {s} 0 $ 0 $ 0 $ 0 $ {这允许初始接口无界或什至不是局部正方形的集成曲率。另外，如果初始曲率是正方形的，则我们获得了最佳速率$ \ frak {s} $的收敛性。
The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where the latter is due to surface tension. To leading order, both the Muskat problems with and without surface tension effect are scaling invariant in the Sobolev space $H^{1+\frac{d}{2}}(\mathbb{R}^d)$, where $d$ is the dimension of the interface. We prove that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with surface tension $\frak{s}$ converge to the unique solution of the Muskat problem without surface tension locally in time with the rate $\sqrt{\frak{s}}$ when $\frak{s}\to 0$. This allows for initial interfaces that have unbounded or even not locally square integrable curvature. If in addition the initial curvature is square integrable, we obtain the convergence with optimal rate $\frak{s}$.
