Local Well-Posedness For Barotropic Compressible Fluid-Viscoelastic Shell Interactions
Pierre Marie Ngougoue Ngougoue
TL;DR
This work establishes local-in-time existence and uniqueness of strong solutions for a 3D isentropic compressible fluid interacting with a boundary-viscoelastic shell in a purely Eulerian framework. It uses a localized Hanzawa transform to straighten the moving interface and decouple the transport and momentum–structure effects: the continuity equation is solved by characteristics in the transformed frame, while the momentum–structure system is handled via linearisation, maximal regularity, and a fixed-point argument on a fixed reference domain. A Banach contraction couples these two steps, providing a short-time, fully nonlinear local well-posedness result with explicit viscosity dependence in the energy estimates, making inviscid limits transparent. The analysis extends strong boundary results to bending shells and complements global finite-energy weak theories for compressible FSI by delivering a robust local strong solution in the same geometric configuration, all within an Eulerian setting.
Abstract
We study a three-dimensional barotropic compressible Navier-Stokes flow interacting with a viscoelastic shell that occupies a portion of the fluid boundary. The analysis is entirely Eulerian and the moving interface is parametrised by a localised Hanzawa transform supported near the shell patch, which preserves the transport structure of the continuity equation and avoids a global Lagrangian map. We prove local-in-time existence and uniqueness of strong solutions for compatible data and without imposing a vanishing initial shell displacement. The proof combines a well-posedness theory for the continuity equation, solved by the method of characteristics in the Hanzawa frame, with an analysis of the momentum-structure subproblem carried out by the classical linearisation-energy estimate -- fixed-point scheme on a fixed reference domain. A Banach fixed point then couples the two steps and closes the argument on a short time interval. We work with a viscosity-weighted energy that makes the scaling in the shear and bulk viscosities explicit. This yields bounds whose constants grow at most linearly in these parameters. In particular, the estimates are inviscid-limit compatible. The result complements the global finite-energy weak theory for compressible fluid-shell interaction by providing a local strong well-posedness statement in the same geometric configuration. It also extends strong boundary results beyond beams and plates to bending shells, and addresses the wave-to-bending direction suggested in the literature on compressible fluid-structure interaction at structural boundaries, while remaining fully Eulerian.