Asymptotics of two-dimensional hydroelastic waves: The zero mass, zero bending limit

TL;DR

This paper investigates the behavior of two-dimensional hydroelastic waves as elastic effects diminish, establishing uniform estimates and showing convergence to vortex sheet solutions with zero elastic parameters.

Contribution

The authors prove the uniform well-posedness and convergence of hydroelastic wave solutions as elastic bending and mass parameters tend to zero.

Findings

Solutions form a Cauchy sequence as parameters vanish

Hydroelastic waves converge to vortex sheet solutions

Uniform estimates are established independent of parameters

Abstract

We consider two-dimensional hydroelastic waves, in which a free fluid surface separates two fluids of infinite vertical extent. Elastic effects are accounted for at the interface, with a parameter measuring the elastic bending force and another parameter measuring the mass of the elastic sheet. In prior work, the authors have demonstrated well-posedness of this initial value problem in Sobolev spaces. We now take the limit as these two parameters vanish. Since the size of the time interval of existence given by this prior theory vanishes as the mass and bending parameters go to zero, we now establish estimates which are uniform with respect to these parameters. We may then make an additional estimate which demonstrates that the solutions form a Cauchy sequence as the parameters go to zero, so that the limit may be taken. This demonstrates that the vortex sheet with surface tension is the zero mass, zero bending limit of hydroelastic waves in two spatial dimensions.