On the existence and the uniqueness of the solution to a fluid-structure interaction problem

Abstract

In this paper we consider the linearized version of a system of partial differential equations arising from a fluid-structure interaction model. We prove the existence and the uniqueness of the solution under natural regularity assumptions.

On a fluid-structure interaction problem for plaque growth

“…Moreover, after linearization one obtains a two-phase Stokes type problem, which ensures us to get the solvabilities and regularities of fluid and solid velocities by maximal regularity theory. In a recent work [9], a similar stress tensor of solid part was also considered to investigate weak solutions of the interaction between an incompressible fluid and an incompressible immersed viscous-hyperelastic solid structure.…”

On the interface matrix for fluid-structure interaction problems with fictitious domain approach

“…In [9], it was shown that a linearized version of Problem 1 admits a unique solution. See also [8] for a review on the state of the art about this problem.…”

“…Starting from here, we then introduced a Lagrange multiplier with the effect that the movement of the structure is managed through a bilinear form that contains the multiplier. As part of our research, several theoretical aspects have been addressed over the years: in [7], we showed the well-posedness of both continuous and discrete problems, in [9] the existence and the uniqueness of the solution were proved in the case of the linearized problem. These results are summarized in a unified setting in [8].…”

Unfitted mixed finite element methods for elliptic interface problems

“…In [6] continuous piecewise bilinear finite element spaces are considered on quadrilateral meshes. Recently, [5] showed, in the framework of FSI, the stability of a linearization of the continuous problem and introduced a unified setting for the choice of the finite element spaces. This setting allows for more general choices of spaces.…”