Abstract
In this paper, we review and comment upon recently derived results for time dependent partial differential equation (PDE) models, which have been used to describe the various fluid-structure interactions which occur in nature. For these fluid-structure PDEs, this survey is particularly focused on the authors' results of (i) semigroup wellposedness, (ii) stability, and (iii) backward uniqueness.
Highlights
In particular: As we shall see, the elimination of the pressure term p(t, x) in (3)-(5) cannot be accomplished by an application of the classic Leray Projector, as is typically done with uncoupled fluid flow PDE models under the so-called “no-slip” boundary condition; the situation calls for a different approach
Pressure p(t) is eliminated in [14] by the means of equating the PDE (3)-(5) below with an appropriate variational relation; on the other hand, in the present survey the pressure term will be eliminated by identifying it as the solution of a certain elliptic boundary value problem, the forcing and boundary terms of which are composed of fluid and structure quantities
We will announce and explain our results in the context of a more canonical fluid-structure model, a model whose relatively simple makeup will allow the reader to quickly digest our posted results without undue frustration at cumbersome notation, we will start by presenting the “physically relevant” PDE model which appears in the aforesaid [27]
Summary
We will announce and explain our results in the context of a more canonical fluid-structure model, a model whose relatively simple makeup will allow the reader to quickly digest our posted results without undue frustration at cumbersome notation, we will start by presenting the “physically relevant” PDE model which appears in the aforesaid [27]. With the geometry {Ωf , Ωs} as described above (and again with unit normal ν exterior to Ωf ), and the stress-strain relations defined in (1)-(2), we are in a position to describe the fluid-structure interactive PDE, which appears in [27]. We note immediately that the fluid component of the initial data is not that which is used in classical Navier-Stokes problems, in which the no-slip boundary condition is in play. In such uncoupled fluid flow problems, a function g is in the (Leray) space of wellposedness if g ∈ N ull(div) and g ·ν = 0 on all of ∂Ωf. We emphasize here, and will throughout, that the results to be announced for the canonical fluidstructure model (9)-(11) below are valid for the Stokes-Lame PDE model (3)-(5)
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