We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension d of the interface. The Muskat problem is scaling invariant in the Sobolev space $$H^{s_c}({\mathbb {R}}^d)$$ H s c ( R d ) where $$s_c=1+\frac{d}{2}$$ s c = 1 + d 2 . Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $$H^s({\mathbb {R}}^d)$$ H s ( R d ) , $$s>s_c$$ s > s c . Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh–Taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid. The starting point of this work is a reformulation solely in terms of the Drichlet–Neumann operator. The key elements of proofs are new paralinearization and contraction results for the Drichlet–Neumann operator in rough domains.

中文翻译：

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马斯卡特问题适定性的超微分方法

我们研究了一种流体或两种流体的 Muskat 问题，有或没有粘度跳跃，有或没有刚性边界，以及界面的任意空间维度 d。Muskat 问题是 Sobolev 空间中的缩放不变性 $$H^{s_c}({\mathbb {R}}^d)$$ H sc ( R d ) 其中 $$s_c=1+\frac{d}{2 }$$ sc = 1 + d 2 。采用超微分方法，我们证明了任何亚临界 Sobolev 空间中大数据的局部适定性 $$H^s({\mathbb {R}}^d)$$ H s ( R d ) , $$s>s_c$ $ s > sc 。此外，刚性边界只需要是 Lipschitz 并且可以有任意大的变化。Rayleigh-Taylor 稳定性条件是在两种具有粘度跳跃的流体的情况下假设的，但证明在一种流体的情况下自动满足。这项工作的出发点是仅根据 Drichlet-Neumann 算子重新表述。证明的关键要素是粗糙域中 Drichlet-Neumann 算子的新的平行线性化和收缩结果。