标题： 适定性与两相周期准静态Stokes流的Rayleigh-Taylor不稳定性 显示英文标题

标题： Well-posedness and Rayleigh-Taylor instability of the two-phase periodic quasistationary Stokes flow

摘要： 我们研究在一般流体（可能具有不同的粘度和密度）背景下由表面张力和重力效应驱动的二维两相、水平周期性准稳态斯托克斯流。假设分隔流体的尖锐界面是周期函数的图像。然后将数学模型重新表述为仅涉及参数化界面函数的完全非线性和非局部演化方程。我们的主要结果包括适定性和抛物线平滑性质，以及在次临界Sobolev空间中平衡解的研究。特别是，我们建立了小指状平衡解的瑞利-泰勒不稳定性，并证明了平坦界面的稳定性特性取决于某个参数的符号。 显示英文摘要

摘要： We study the two-phase, horizontally periodic, quasistationary Stokes flow in two dimensions driven by surface tension and gravity effects in the general context of fluids with (possibly) different viscosities and densities. The sharp interface which separates the fluids is assumed to be the graph of a periodic function. The mathematical model is then recast as a fully nonlinear and nonlocal evolution equation involving only the function parametrizing the interface. Our main results include well-posedness and a parabolic smoothing property, as well as a study of equilibrium solutions in subcritical Sobolev spaces. In particular, we establish the Rayleigh-Taylor instability of small, finger-shaped equilibria and prove that the stability properties of flat interfaces depend on the sign of a certain parameter.