Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

Abstract

We study the well-posedness of a total linearization, with respect to a perturbation of the external forcing, of a free-boundary nonlinear elasticity-incompressible fluid interaction. The total linearization for the coupling modeled by the Navier-Stokes equations and the nonlinear equations of elastodynamics was obtained recently in [L. Bociu and J.-P. Zolésio, Evol. Equ. Control Theory, 2 (2013), pp. 55-79]. The equations and the free boundary were linearized together, and the result turned out to be quite different from the usual coupling of classical linear models. New additional terms are present on the common interface, some of them involving boundary curvatures and boundary acceleration. These terms play an important role in the final linearized system and cannot be neglected; their presence also introduces new challenges in the well-posedness analysis, which proceeds to establish that the evolution operator associated to the linearized system can be represented as a bounded perturbation of a maximal dissipative semigroup generator. Introduction.A typical fluid-structure interaction system is composed of a solid either surrounding a fluid flow (arteries, pipelines, reservoir tanks, etc.) or surrounded by a fluid or gas (blood cells, marine animals, bridge supports, aircraft, automobiles, etc.). This model falls in the category of free boundary problems, since the motion of the common boundary between the solid and the fluid is one of the unknowns in the coupled system. The ubiquity of this interaction type has led to a rapidly growing research interest in this model. Its mathematical solvability [5,9,15,17,22,21,26,32,33,34,37,38,49,55,52], numerical approximations [23,29,39,54,53,44,46], and stability [14,27,28,31] have been intensively studied.Mathematically, this interaction is described by a partial differential equation (PDE) system that couples the parabolic (fluid) and hyperbolic (elasticity) phases, where the key issue is that the traces of the elastic component at the energy level are not defined via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently.The first major development appeared in [16], where the authors established local existence and regularity of solutions for the coupling of Navier-Stokes equations and a linear elasticity model, describing a body submerged in a fluid, under the assumptions of smooth initial data; specifically, the initial fluid velocity w 0 has Sobolev regularity H 5 , and the initial data for the elastic component (ϕ 0 , ϕ 1 ) belongs to H 3 × H 2 . Moreover, due to the incompressibility condition on the fluid, the uniqueness result

A uniform discrete inf-sup inequality for finite element hydro-elastic models

“…research efforts continue to pursue linearized systems as well; among the relatively more recent contributions are [1,3,4,19,14,2,12].…”

“…We stress again that in the definition of H f l the zero normal component condition is enforced only on the exterior non-slip boundary Γ f , but not on the entire boundary of the fluid domain ∂Ω f . The semigroup well-posedness of (1)-(3) was discussed at length in [1,3,4] (see also [12] for the semigroup result on a related, but more complex model linearized around a steady regime).…”

Linearized hydro-elasticity: A numerical study

“…The well-posedness analysis for the new linearization [7] was recently addressed in [5]. The "new" first-order terms (7) present in the Neumann boundary condition for the elastic solid result in an oblique derivative problem.…”

“…Hence the additional terms appearing in the linearization cannot be trivially handled as mere perturbations of the elastic component even if we impose a size condition on them; with the smallness assumption in place the regularity issues still have to be addressed. In [5] the well-posedness of this model is investigated via semigroup theory in the special case when the linearization is performed near a sufficiently smooth low-velocity steady-state regime. The maximality result for the evolution generator was demonstrated using a variational approach and the Babuška-Brezzi theorem following the strategy in [1].…”

“…Of course, the gain partly follows from the assertion that linearization is performed around a smooth solution to the nonlinear problem in the first place, such as linearization near equilibrium. The analysis in [5], however, shows that if the linearization takes into account geometry of the domain and is performed near a nonzero steady regime then existence of finite-energy solutions likely requires a sufficiently large viscosity-this aspect was irrelevant in the simple classical Stokes-elasticity coupling. The analytic study in [5] does not not provide precise quantitative estimates on the relative parameter sizes, and the current paper aims at investigating this aspect further.…”