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 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.
Highlights
This work is stimulated by the physical models studied in [12,13], where longitudinal elastic waves of a membrane are coupled to viscous fluid flow in the enclosing half space
The aims are to understand the damping of the elastic waves through the coupling to the viscous fluid, on the one hand, and to explain the appearance of the non-classical dispersion relation, on the other hand
We have shown that the fractionally damped wave equation can be obtained as a scaling limit from a bulk-interface coupling between a wave equation for a membrane and a viscous fluid motion in the adjacent half space
Summary
This work is stimulated by the physical models studied in [12,13], where longitudinal elastic waves of a membrane are coupled to viscous fluid flow in the enclosing half space. The aims are to understand the damping of the elastic waves through the coupling to the viscous fluid, on the one hand, and to explain the appearance of the non-classical dispersion relation, on the other hand. Mathematics Subject Classification: 35Q35, 35Q74, 74J15 Keywords: Bulk-interface coupling, Surface waves, Energy–dissipation balance, Fractional derivatives, Parabolic Dirichlet-to-Neumann map, Convergence of semigroups. By starting from the natural energy and dissipation in the PDE system (1.4) with ε = 0 and the explicit solution for v(t, x, z) in terms of U (τ, x), we obtain a natural energy functional E for the fractionally damped wave equation that is non-local in time: E(U (t), U (·) [0,t]) =.
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