661 research outputs found
Flow-plate interactions: Well-posedness and long-time behavior
We consider flow-structure interactions modeled by a modified wave equation
coupled at an interface with equations of nonlinear elasticity. Both subsonic
and supersonic flow velocities are treated with Neumann type flow conditions,
and a novel treatment of the so called Kutta-Joukowsky flow conditions are
given in the subsonic case. The goal of the paper is threefold: (i) to provide
an accurate review of recent results on existence, uniqueness, and stability of
weak solutions, (ii) to present a construction of finite dimensional,
attracting sets corresponding to the structural dynamics and discuss
convergence of trajectories, and (iii) to state several open questions
associated with the topic. This second task is based on a decoupling technique
which reduces the analysis of the full flow-structure system to a PDE system
with delay.Comment: 1 figure. arXiv admin note: text overlap with arXiv:1208.5245,
arXiv:1311.124
Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary
We address semigroup well-posedness of the fluid-structure interaction of a
linearized compressible, viscous fluid and an elastic plate (in the absence of
rotational inertia). Unlike existing work in the literature, we linearize the
compressible Navier-Stokes equations about an arbitrary state (assuming the
fluid is barotropic), and so the fluid PDE component of the interaction will
generally include a nontrivial ambient flow profile . The
appearance of this term introduces new challenges at the level of the
stationary problem. In addition, the boundary of the fluid domain is
unavoidably Lipschitz, and so the well-posedness argument takes into account
the technical issues associated with obtaining necessary boundary trace and
elliptic regularity estimates. Much of the previous work on flow-plate models
was done via Galerkin-type constructions after obtaining good a priori
estimates on solutions (specifically \cite {Chu2013-comp}---the work most
pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach,
with a view of associating solutions of the fluid-structure dynamics with a
-semigroup on the natural
finite energy space of initial data. So, given this approach, the major
challenge in our work becomes establishing of the maximality of the operator
which models the fluid-structure dynamics. In sum: our main
result is semigroup well-posedness for the fully coupled fluid-structure
dynamics, under the assumption that the ambient flow field has zero normal component trace on the boundary (a
standard assumption with respect to the literature). In the final sections we
address well-posedness of the system in the presence of the von Karman plate
nonlinearity, as well as the stationary problem associated with the dynamics.Comment: 1 figur
Eliminating flutter for clamped von Karman plates immersed in subsonic flows
We address the long-time behavior of a non-rotational von Karman plate in an
inviscid potential flow. The model arises in aeroelasticity and models the
interaction between a thin, nonlinear panel and a flow of gas in which it is
immersed [6, 21, 23]. Recent results in [16, 18] show that the plate component
of the dynamics (in the presence of a physical plate nonlinearity) converge to
a global compact attracting set of finite dimension; these results were
obtained in the absence of mechanical damping of any type. Here we show that,
by incorporating mechanical damping the full flow-plate system, full
trajectories---both plate and flow---converge strongly to (the set of)
stationary states. Weak convergence results require "minimal" interior damping,
and strong convergence of the dynamics are shown with sufficiently large
damping. We require the existence of a "good" energy balance equation, which is
only available when the flows are subsonic. Our proof is based on first showing
the convergence properties for regular solutions, which in turn requires
propagation of initial regularity on the infinite horizon. Then, we utilize the
exponential decay of the difference of two plate trajectories to show that full
flow-plate trajectories are uniform-in-time Hadamard continuous. This allows us
to pass convergence properties of smooth initial data to finite energy type
initial data. Physically, our results imply that flutter (a non-static end
behavior) does not occur in subsonic dynamics. While such results were known
for rotational (compact/regular) plate dynamics [14] (and references therein),
the result presented herein is the first such result obtained for
non-regularized---the most physically relevant---models
Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions
We study the finite-horizon optimal control problem with quadratic
functionals for an established fluid-structure interaction model. The coupled
PDE system under investigation comprises a parabolic (the fluid) and a
hyperbolic (the solid) dynamics; the coupling occurs at the interface between
the regions occupied by the fluid and the solid. We establish several trace
regularity results for the fluid component of the system, which are then
applied to show well-posedness of the Differential Riccati Equations arising in
the optimization problem. This yields the feedback synthesis of the unique
optimal control, under a very weak constraint on the observation operator; in
particular, the present analysis allows general functionals, such as the
integral of the natural energy of the physical system. Furthermore, this work
confirms that the theory developed in Acquistapace et al. [Adv. Differential
Equations, 2005] -- crucially utilized here -- encompasses widely differing PDE
problems, from thermoelastic systems to models of acoustic-structure and, now,
fluid-structure interactions.Comment: 22 pages, submitted; v2: misprints corrected, a remark added in
section
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