Abstract

An initial boundary value problem with a free boundary for a third-order integro-differential equation for the unsteady flow of an aqueous polymer solution in a strip is studied. Questions concerning the solvability of the problem and the qualitative behavior of the solution in dependence on the initial data are investigated: time-local resolvability is established under natural conditions on the input data and conditions of global solvability are found for the problem of strip narrowing. Conditions for the bowing-up of the solution in finite time in the model problem of strip expansion are presented. The problem of small perturbations of the flow of a strip of viscous fluid with respect to a parameter proportional to the relaxation viscosity is also consider.

中文翻译：

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聚合物溶液模型中的自由边界问题

摘要

研究了带状聚合物水溶液不稳定流动的三阶积分微分方程具有自由边界的初始边界值问题。研究了有关问题的可解性和溶液的定性行为（取决于初始数据）的问题：在自然条件下根据输入数据建立了时局部可解性，并为带材变窄问题找到了整体可解性的条件。给出了带钢扩展模型问题中有限时间内翘曲的条件。还考虑了粘性流体带的流动相对于与驰豫粘度成比例的参数的小扰动的问题。