We prove that the quadratic approximation of the capillarity-driven free-boundary Darcy flow, derived in Granero-Belinchón and Scrobogna (2019), is well posed in ${\dot{H}}^{3∕2}\left({\mathbb{S}}^1\right)$, and globally well-posed if the initial datum is small in ${\dot{H}}^{3∕2}\left({\mathbb{S}}^1\right)$.

中文翻译：

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临界Sobolev空间中多孔介质中毛细驱动的自由边界达西流动的渐近模型的适定性

我们证明，由Granero-Belinchón和Scrobogna（2019）推导的毛细作用驱动的自由边界Darcy流的二次逼近很好地提出于 ${\dot{H}}^{3∕2}\left({\mathbb{小号}}^{\mathrm{1个}}\right)$，并且如果初始基准面较小， ${\dot{H}}^{3∕2}\left({\mathbb{小号}}^{\mathrm{1个}}\right)$。