Abstract

We consider the dynamics of finite perturbations of a plane phase transition surface in the problem of evaporation of a fluid inside a low-permeability layer of a porous medium. In the case of a nonwettable porous medium, the problem has two stationary solutions, each containing a discontinuity. These discontinuities correspond to plane stationary phase transition surfaces located inside the low-permeability porous layer. One of these surfaces is unstable with respect to long-wavelength perturbations, while the other is stable. We study the evolution of perturbations of the stable plane phase transition surface. It is known that when two phase transition surfaces are located close enough to each other, the dynamics of a weakly nonlinear and weakly unstable wave packet is described by the Kolmogorov–Petrovskii–Piskunov (KPP) diffusion equation. As traveling wave solutions, this equation has heteroclinic solutions with either oscillating or monotonic structure of the front. The boundary value problem in the full statement, which should be considered if the distance between the stable and unstable plane phase transition surfaces is not small, also has similar solutions. We formulate a sufficient condition for the decrease of finite perturbations of the stable plane phase transition surface. This condition depends on their position with respect to the standing wave type and traveling front type solutions of the model equations in the model description when the KPP equation holds.

中文翻译：

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多孔介质中扩散作用下的扰动动力学

摘要

在多孔介质的低渗透层内部的流体蒸发问题中，我们考虑了平面相变表面的有限摄动动力学。对于不可润湿的多孔介质，问题有两个固定解，每个固定解都包含一个不连续性。这些不连续对应于位于低渗透性多孔层内部的平面固定相变表面。这些表面中的一个相对于长波长扰动是不稳定的，而另一个则是稳定的。我们研究了稳定平面相变表面的扰动演化。众所周知，当两个相变表面彼此足够靠近时，弱非线性和弱不稳定波包的动力学由Kolmogorov-Petrovskii-Piskunov（KPP）扩散方程描述。作为行波解，该方程具有杂波解决方案，其前部具有振荡或单调结构。如果稳定和不稳定平面相变表面之间的距离不小，则应考虑完整语句中的边值问题，该问题也具有类似的解决方案。我们为减少稳定平面相变表面的有限扰动制定了充分条件。当KPP方程成立时，该条件取决于它们在模型描述中相对于模型方程的驻波类型和行进前型解的位置。如果稳定和不稳定平面相变表面之间的距离不小，也应考虑类似的解决方案。我们为减少稳定平面相变表面的有限扰动制定了充分条件。当KPP方程成立时，该条件取决于它们在模型描述中相对于模型方程的驻波类型和行进前型解的位置。如果稳定和不稳定平面相变表面之间的距离不小，也应考虑类似的解决方案。我们为减少稳定平面相变表面的有限扰动制定了充分条件。当KPP方程成立时，该条件取决于它们在模型描述中相对于模型方程的驻波类型和行进前型解的位置。