具有非线性扩散的薄膜方程自由边界传播：最优结果的尺度一致方法
Free Boundary Propagation of a Thin-Film Equation with Nonlinear Diffusion: Scale-Consistent Methods for Optimal Results
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中文总结 AI 辅助
该论文研究薄膜方程自由边界传播，通过尺度分析获得传播速率最优上下界，揭示四阶与二阶算子在短长期的主导转换及边界不推进条件。
中文摘要 AI 辅助
我们研究了一个同时包含四阶和二阶非线性的退化非线性抛物问题的自由边界传播，该问题可用于模拟弱滑移区域中的粘性薄膜流动，并附加了由分子间力引起的奇异分离压力的贡献。通过仔细处理非线性的各自尺度行为，我们获得了向前支撑传播速率的渐近最优上下界。这些速率表明，四阶算子在短时间尺度上主导传播速率，但长期传播速率由二阶算子控制。我们进一步估计了这种传播速率权衡发生的时间。最后，我们检验了确保自由边界在给定方向上不立即推进的充分条件，从而详细描绘了二阶/四阶算子之间的竞争如何影响解的自由边界动力学。
英文摘要
We investigate the free boundary propagation of a non-linear degenerate parabolic problem with both fourth and second order non-linearities, which may be used to model viscous thin-film flow in the regime of weak slippage with an additional contributions from a singular disjoining pressure brought forth by inter-molecular forces. By carefully treating the individual scaling behaviors of the non-linearities, we obtain asymptotically optimal upper and lower bounds on the rate of forward support propagation. These rates show that the fourth-order operator dominates the propagation rate on short time scales, but the long-term rate of propagation is controlled by the second-order operator. We estimate further the time at which this trade-off in propagation rate occurs. Lastly, we examine sufficient conditions to ensure that the free boundary does not immediately advance in a given direction, leading to a detailed picture of how the competition between the second/fourth-order operators influence the free boundary dynamics of solutions.
发表机构
- FAU Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）
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