Mielke, Alexander and Netz, Roland R. and Zendehroud, Sina (2020) A rigorous derivation and energetics of a wave equation with fractional damping. arXiv . pp. 120. (Submitted)

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Official URL: https://arxiv.org/abs/2004.11830
Abstract
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 waterair 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 fractionallydamped 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 energydissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionallydamped wave equation with a time derivative of order 3/2. Keywords: bulkinterface coupling, surface waves, energydissipation balance, fractional derivatives, convergence of semigroups, parabolic DirichlettoNeumann map. MOS: 35Q35 35Q74 74J15
Item Type:  Article 

Subjects:  Mathematical and Computer Sciences > Mathematics > Applied Mathematics 
Divisions:  Department of Mathematics and Computer Science > Institute of Mathematics 
ID Code:  2442 
Deposited By:  Monika Drueck 
Deposited On:  28 May 2020 10:39 
Last Modified:  28 May 2020 10:39 
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