论文标题
亚临界$ l_p $ -sobolev空间中的Muskat问题的适应性
Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
论文作者
论文摘要
我们研究了Muskat问题，这些问题描述了在$ l_p $ setting的二维均匀多孔培养基中的垂直运动，并用$ p \ in（1，\ infty）$。 Sobolev space $ w^s_p（\ Mathbb {r}）$，$ s = 1+1/p $是解决此问题的关键空间。我们证明，对于$ s \ in（1+1/p，2），$ rayleigh-taylor条件标识了$ w^s_p（\ mathbb {r}）$的开放子集，其中muskat问题是抛物线词类型。这使我们能够在所有这些亚临界空间以及抛物线平滑属性中建立问题的本地良好性。
We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W^s_p(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.
