[Paper Review] Improved boundary regularity for a Stokes-Lam\\'e system

Francesca Bucci|arXiv (Cornell University)|Sep 10, 2020

Stability and Controllability of Differential Equations37 references43 citations

TL;DR

The paper analyzes a damped Stokes-Lamé fluid–structure interaction, proving boundary trace regularity for the fluid on the interface and enabling infinite-horizon linear-quadratic control via semigroup/interpolation methods.

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 -- in time and space, of local and global nature -- are geared toward ensuring the well-posedness of the algebraic Riccati equations which arise from the associated optimal boundary control problems on an infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.

Motivation & Objective

• Motivate the study of fluid–elastic interface interactions in a fixed, damped setting.
• Establish trace regularity of fluid interface traces necessary for control design.
• Develop an abstract semigroup framework to reformulate the PDE system.
• Enable solvability of algebraic Riccati equations arising from infinite-horizon optimal control.

Proposed method

• Formulate the coupled PDE system as an abstract Cauchy problem y' = Ay + Bg in a suitable Banach space.
• Show the generator A defines an analytic semigroup and the control operator B is admissible.
• Derive trace regularity results for the fluid velocity and its boundary traces on the interface via semigroup/interpolation techniques.
• Use regularity of the elastic stress on the interface (Proposition 2.4) to control boundary terms.
• Apply the infinite-horizon linear-quadratic framework to the damped FSI and connect traceRegularity to Riccati equation well-posedness.

Experimental results

Research questions

• RQ1 What regularity can be obtained for the trace of the fluid velocity u on the interface Γs, and its time derivative u_t, given the damped FSI system?
• RQ2 How does the boundary dissipation (a2 > 0) change stability and regularity compared to the undamped case (a2 = 0)?
• RQ3 Can the established trace regularity support the well-posedness of the associated infinite-horizon linear-quadratic control problem and the corresponding Riccati equations?

Key findings

• The fluid velocity on the interface Γs can be decomposed as u(t) = u1(t) + u2(t), with u1(t) exhibiting exponential decay tempered by a time singularity (t^{-1/4-δ}).
• The second component u2 satisfies Lp-type regularity on the interface, yielding u2|Γs ∈ L^p(0,T; L^2(Γs)) for all p ≥ 1 and finite T.
• Stronger regularity is obtained under higher-regularity initial data: u2|Γs ∈ H^{1+ε/2, 1/2+ε/4}(Σs) and hence u2|Γs ∈ C([0,T], L^2(Γs)).
• The fluid acceleration trace u_t on the interface belongs to L^q(0,T; H^{1/2−θ−δ}(Γs)) for q < 2/(2−δ), and in particular lies in L^q(0,T; L^2(Γs)) for q < 4/(3+2θ).
• These trace regularity results are obtained without smoothing observations and are compatible with full quadratic energy functionals, enabling application of the infinite-horizon LQ theory.