# American Institute of Mathematical Sciences

eISSN:
2163-2480

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## Evolution Equations & Control Theory

December 2016 , Volume 5 , Issue 4

Special issue on fluid-structure interactions

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2016, 5(4): i-iii doi: 10.3934/eect.201604i +[Abstract](1399) +[PDF](110.5KB)
Abstract:
This special volume of Evolution Equations and Control Theory commemorates the results presented at a mini-symposium on Analysis and Control of Fluid Models and Flow-coupled Systems" in December 2015. This meeting was part of the SIAM Conference on Analysis of Partial Differential Equations, held December 7--10, 2015 in Scottsdale, Arizona at the DoubleTree Resort by Hilton in Paradise Valley. The mini-symposium was organized by the Editors: Marcelo Disconzi, Irena Lasiecka, Daniel Toundykov, and Justin Webster.

2016, 5(4): 475-487 doi: 10.3934/eect.2016015 +[Abstract](1968) +[PDF](369.9KB)
Abstract:
Two variations of a basic model for a cochlea are described which consist of a basilar membrane coupled with a linear potential fluid. The basilar membrane is modeled as an array of oscillators which may or may not include longitudinal elasticity. The fluid is assumed to be a linear potential fluid described by Laplace's equation in a domain that surrounds the basilar membrane. The problem of controllability of the system is considered with control active on a portion of the basilar membrane. Approximate controllability is proved for both models and moreover lack of exact controllability is shown to hold when longitudinal stiffness is not included.
2016, 5(4): 489-514 doi: 10.3934/eect.2016016 +[Abstract](2026) +[PDF](561.2KB)
Abstract:
The purpose of this paper is to complement available literature on sharp regularity theory of second order mixed hyperbolic problem of Neumann type [13,15,26] with a series of new results in the case--so far rather unexplored--where the Neumann boundary term (input, control) possesses a regularity below $L^2(\Gamma)$ in space on the boundary $\Gamma$. We concentrate on the cases $H^{-\frac{1}{2}}(\Gamma))$, $H^{-\beta}(\Gamma))$, $H^{-1}(\Gamma))$, $\beta$ being a distinguished parameter of the problem. Our present results are consistent with the sharp result of [13,15,26] (obtained through a pseudo-differential/micro-local analysis approach), whose philosophy is expressed by a gain of $\beta$ in space regularity in going from the boundary control to the position in the interior. A number of physically relevant illustrations are given.
2016, 5(4): 515-531 doi: 10.3934/eect.2016017 +[Abstract](1829) +[PDF](474.8KB)
Abstract:
A seminal result concerning finite element (FEM) approximations of the Stokes equation was the discrete inf-sup inequality that is uniform with respect to the mesh size parameter. This inequality leads to optimal error estimates for the FEM scheme. The original version pertains to the Stokes system with non-slip boundary condition on the entire boundary. On the other hand, in fluid-structure interaction problems, the interface dynamics between the fluid and the solid satisfies velocity and stress matching constraints. As a result, the pressure variable is no longer determined up to a constant and becomes subject to non-homogeneous Dirichlet conditions on the common interface. In this context, we establish a uniform discrete inf-sup estimate for a fluid-structure FEM implementation based on Taylor-Hood elements, and use this inequality to verify some stability and error estimates of this numerical scheme. An added benefit of this framework is that it does not require the Poisson-equation approach to solve for the pressure variable.
2016, 5(4): 533-559 doi: 10.3934/eect.2016018 +[Abstract](1794) +[PDF](1230.2KB)
Abstract:
In view of control and stability theory, a recently obtained linearization around a steady state of a fluid-structure interaction is considered. The linearization was performed with respect to an external forcing term and was derived in an earlier paper via shape optimization techniques. In contrast to other approaches, like transporting to a fixed reference configuration, or using transpiration techniques, the shape optimization route is most suited to incorporating the geometry of the problem into the analysis. This refined description brings up new terms---missing in the classical coupling of linear Stokes flow and linear elasticity---in the matching of the normal stresses and the velocities on the interface. Later, it was demonstrated that this linear PDE system generates a $C_0$ semigroup, however, unlike in the standard Stokes-elasticity coupling, the wellposedness result depended on the fluid's viscosity and the new boundary terms which, among other things, involve the curvature of the interface. Here, we implement a finite element scheme for approximating solutions of this fluid-elasticity dynamics and numerically investigate the dependence of the discretized model on the new" terms present therein, in contrast with the classical Stokes-linear elasticity system.
2016, 5(4): 561-566 doi: 10.3934/eect.2016019 +[Abstract](1501) +[PDF](299.1KB)
Abstract:
We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.
2016, 5(4): 567-603 doi: 10.3934/eect.2016020 +[Abstract](1918) +[PDF](974.2KB)
Abstract:
We consider a nonlinear (Berger or Von Karman) clamped plate model with a piston-theoretic right hand side---which includes non-dissipative, non-conservative lower order terms. The model arises in aeroelasticity when a panel is immersed in a high velocity linear potential flow; in this case the effect of the flow can be captured by a dynamic pressure term written in terms of the material derivative of the plate's displacement. The effect of fully-supported internal damping is studied for both Berger and von Karman dynamics. The non-dissipative nature of the dynamics preclude the use of strong tools such as backward-in-time smallness of velocities and finiteness of the dissipation integral. Modern quasi-stability techniques are utilized to show the existence of compact global attractors and generalized fractal exponential attractors. Specific results here depend on the size of the damping parameter and the nonlinearity in force. For the Berger plate, in the presence of large damping, the existence of a proper global attractor (whose fractal dimension is finite in the state space) is shown via a decomposition of the nonlinear dynamics. This leads to the construction of a compact set upon which quasi-stability theory can be implemented. Numerical investigations for appropriate 1-D models are presented which explore and support the abstract results presented herein.
2016, 5(4): 605-629 doi: 10.3934/eect.2016021 +[Abstract](1782) +[PDF](544.0KB)
Abstract:
We study dynamics of a coupled system consisting of the 3D Navier--Stokes equations which is linearized near a certain Poiseuille type flow between two unbounded circular cylinders and nonlinear elasticity equations for the transversal displacements of the bounding cylindrical shells. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.
2016, 5(4): 631-645 doi: 10.3934/eect.2016022 +[Abstract](1676) +[PDF](440.2KB)
Abstract:
Of concern is the motion of two fluids separated by a free interface in a porous medium, where the velocities are given by Darcy's law. We consider the case with and without phase transition. It is shown that the resulting models can be understood as purely geometric evolution laws, where the motion of the separating interface depends in a non-local way on the mean curvature. It turns out that the models are volume preserving and surface area reducing, the latter property giving rise to a Lyapunov function. We show well-posedness of the models, characterize all equilibria, and study the dynamic stability of the equilibria. Lastly, we show that solutions which do not develop singularities exist globally and converge exponentially fast to an equilibrium.
2016, 5(4): 647-659 doi: 10.3934/eect.2016023 +[Abstract](2343) +[PDF](406.6KB)
Abstract:
We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha>0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$.
2016, 5(4): 661-676 doi: 10.3934/eect.2016024 +[Abstract](1783) +[PDF](422.8KB)
Abstract:
A third order in time nonlinear equation is considered. This particular model is motivated by High Frequency Ultra Sound (HFU) technology which accounts for thermal and molecular relaxation. The resulting equations give rise to a quasilinear-like evolution with a potentially degenerate damping [23]. The purpose of this paper is twofold: (1) to provide a brief review of recent results in the area of long time behavior of solutions to of MGT equation, (2) to provide recent results pertaining to decay of energy associated with the model accounting for molecular relaxation which is locally distributed.

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