Improved boundary regularity for a Stokes-Lamé system

Abstract

This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the interface between the body and the fluid is established, in the case a suitable boundary dissipation is present. These regularity estimates are geared toward ensuring the well-posedness of the Riccati equations which arise from the associated optimal boundary control problems on a finite as well as infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.

On the infinitesimal generator of an optimal state semigroup

“…Theorem 1.9, shows that this is indeed the case, provided that the parameters γ and ǫ which occur in the Assumptions 1.4 fulfil ǫ < 1 − γ. This relation is feasible: to support the assertion, we revisit the actual values of γ and ǫ brought about by the (model-specific) trace regularity results established in [13], [2], [11], [12] in regard to distinct PDE problems, in the process of establishing that these fall into the present underlying framework.…”

“…We recall and discuss briefly three distinct PDE problems describing distinct physical interactions: from mechanical-thermal and acoustic-structure ones (in 2D and 3D, respectively) to fluid-elasticity ones in 3D. Each of these illustrations has been previously shown to fall within the class of abstract control systems defined by the Assumptions 1.4; see the papers [2], [11], [12], respectively. The values of γ and ǫ specifically suited for either case had explicitly arisen during the proofs of the soughtafter (respective) trace regularity results that pertain to a certain component of the solution to the uncontrolled PDE problem.…”

“…The PDE system in (3.3) shares with the previous illustrations the features that it comprises both parabolic and hyperbolic equations, and moreover the parabolic component is subjected to a boundary control action. Notably, its abstract reformulation falls into the framework characterized by the Assumptions 1.4, as proved in [12].…”

“…The functional setup for the IBVP (3.3) follows -in its basic elements -the one introduced in [8] for the true nonlinear problem. Also, the perspective taken in [12] to attain the sought regularity results is akin to the one adopted in [20], [14] and [15] in the study of the optimal boundary control problem associated with its (undamped) linearization. We note that the regularity analysis that has been carried out in the aforementioned works benefited from the enhanced regularity of the boundary traces of the normal component of the stress tensor on the interface, valid for weak as well as semigroup solutions.…”

“…where H denotes the (null-div) space pertaining to the fluid component, that is 3 is the natural energy space for the Lamé system. As before, the IBVP (3.3) can be recast as the abstract Cauchy problem (1.3), having set y(t) := y(t, •) ≡ (u(t, •), w(t, •), w t (t, •)) , g(t) = g(t, •) , and with suitable operators A and B explicitly identified; see [12,Section 2.1]. The abstract setup introduced in [12] has proved well-suited for the derivation of the boundary regularity estimates that allow for the invocation of the theory of the infinite time horizon optimal control problem devised in [4].…”