2020/04/24 by Alexander Mielke, Roland R. Netz, Mielke, Alexander +3
Engineering · Mathematics · Physics and Astronomy · #35Q74 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Soft Condensed Matter (cond-mat.soft) #Stability and Controllability of Differential Equations #cond-mat.soft #math-ph #math.AP #math.MP #msc:35Q74
paper · pdf · doi:10.48550/arxiv.2004.11830
arxiv created 2020/04/24 · openalex publication_date 2020/04/24 · arxiv updated 2020/04/27 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
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