论文标题
波动方程的严格衍生和能量分数阻尼
A rigorous derivation and energetics of a wave equation with fractional damping
论文作者
论文摘要
我们考虑了一个线性系统，该系统由水平高表面上的线性波方程组成，以及下面半空间中的抛物线方程。该模型描述了水气界面上有机单层中的纵向弹性波，这是一种实验设置，与理解生物膜中的波传播有关。我们研究了相关水平长度尺度比垂直长度尺度大得多的缩放模式，并提供了严格的极限，从而导致膜的分数抑制波方程。我们通过线性半群理论提供相关的存在结果，并在缩放限制中显示解决方案的收敛性。此外，基于完整模型的能量驱动结构，我们得出了一个自然能量和分数抑制波方程的自然耗散函数，并具有3/2阶的时间导数
We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2
