Abstract We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase.

中文翻译：

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伪微分算子的半经典估计和不稳定状态下的 Muskat 问题

摘要 我们获得了具有低正则符号的伪微分算子的新半经典估计。此类符号自然出现在与最近通过凸积分构造的不稳定 Muskat 问题的弱解相关的柯西问题中。特别是，我们的新估计揭示了混合区打开速度与界面规律之间的密切关系。