A wave equation on a time-dependent domain is considered. The shape of the domain changes according to a prescribed space/time-dependent velocity field. On the moving boundary the solution satisfies zero Dirichlet condition. It is known that if the domain keeps expanding at a “subsonic” speed, then the associated finite energy decays uniformly. Here, the scenario of interest is when the domain remains bounded and undergoes phases of expansion and contraction. Although the energy identity in this case is not necessarily conservative, it is shown that the L2 space–time norm of the normal trace remains a priori bounded at small normal speeds of the boundary, analogously to the classical Dirichlet wave problem on a static domain. In addition, it is demonstrated that small normal velocity but very large acceleration of the boundary is compatible with the known existence theory, provided the magnitude of the deformations is relatively small. An “adaptive” boundary movement control is proposed and implemented numerically. The control action is dynamically computed from the normal trace data and dissipates the energy by means of small deformations of the domain only.

We consider a cantilevered (clamped-free) beam in an axial potential flow. Certain flow velocities may bring about a bounded-response instability in the structure, termed flutter. As a preliminary analysis, we employ the theory of large deflections and utilize a piston-theoretic approximation of the flow for appropriate parameters, yielding a nonlinear (Berger/WoinowskyKrieger) beam equation with a nondissipative right-hand side. As we obtain this structural model via a simplification, we arrive at a nonstandard nonlinear boundary condition that necessitates careful well-posedness analysis. We account for beam rotational inertia effects and discuss technical issues that necessitate this feature. We demonstrate nonlinear semigroup well-posedness of the model with the rotational inertia terms. For the case with no rotational inertia, we utilize a Galerkin approach to establish the existence of weak, possibly nonunique, solutions. For the former, inertial model, we prove that the associated nongradient dynamical system has a compact global attractor. Finally, we study stability regimes and postflutter dynamics (nonstationary end behaviors) using numerical methods for models with, and without, the rotational inertia terms.

It is known that the linear Stokes-Lame system can be stabilized by a boundary feedback in the form of a dissipative velocity matching on the common interface [5]. Here we consider feedback stabilization for a generalized linear fluid-elasticity interaction, where the matching conditions on the interface incorporate the curvature of the common boundary and thus take into account the geometry of the problem. Such a coupled system is semigroup well-posed on the natural finite energy space [13], however, the system is not dissipative to begin with, which represents a key departure from the feedback control analysis in [5]. We prove that a damped version of the general linear hydro-elasticity model is exponentially stable. First, such a result is given for boundary dissipation of the form used in [5]. This proof resolves a more complex version, compared to the classical case, of the weighted energy methods, and addresses the lack of over-determination in the associated unique continuation result. The second theorem demonstrates how assumptions can be relaxed if a viscous damping is added in the interior of the solid.

Hyporheic flow and nutrient turnover in hyporheic systems are strongly influenced by in-stream bedforms. An accurate representation of topographical variations of the stream-streambed interface is therefore essential in analytical models in order to represent the couplings between hydrological and biogeochemical processes correctly. The classical Toth approach replaces the streambed surface topography by a flat surface which is identical to a truncation of the original physical flow domain into a rectangle. This simplification can lead to biased estimates of hyporheic flow and nutrient cycling within hyporheic systems. We present an alternative analytical modeling approach for solving hyporheic problems without domain truncation that explicitly accounts for topographical variations of the streambed. The presented approach is based on the application of perturbation theory. Applications of the method to hyporheic systems, ranging from the centimeter-scale of rippled bedforms to riffle structures of 10 m and larger scale, indicate a high accuracy of the approach.

This study addresses well-posedness of a Mindlin–Timoshenko (MT) plate model that incorporates nonlinear viscous damping and nonlinear source term in Neumann boundary conditions. The main results verify local and global existence of solutions as well as their continuous dependence on the initial data in appropriate function spaces. Along with (Pei et al. in J Math Anal Appl 418(2):535–568, 2014, in Nonlinear Anal 105:62–85, 2014) this work completes the fundamental well-posedness theory for MT plates under the interplay of damping and source terms acting either in the interior or on the boundary of the plate.

We consider a singular integral operator as a natural generalization to the biharmonic operator that arises in thin plate theory. The operator is built in the nonlocal calculus framework defined in (Math Models Methods Appl Sci 23(03):493–540, 2013) and connects with the recent theory of peridynamics. This framework enables us to consider non-smooth approximations to fourth-order elliptic boundary-value problems. For these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well-defined even for irregular domains. We demonstrate the existence and uniqueness of solutions to these nonlocal problems and demonstrate their L 2-strong convergence to functions in W 2,2 as the nonlocal interaction horizon goes to zero. For regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary-value problems. As a part of our proof we also establish that the nonlocal Laplacian of a smooth function is Lipschitz continuous.

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 bene fit of this framework is that it does not require the Poisson-equation approach to solve for the pressure variable.

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.

The exact controllability problem for several semilinear thin plate models is considered. A distributed control affects a collar of the plate's boundary. The result is semiglobal in the sense that there is no restriction on the size of the initial and the target states, but the controllability time is uniform only with respect to a given bounded set containing these states. Both (i) closed-loop-based and (ii) “pure” open-loop constructions are discussed. Strategy (i) describes exact controls for an abstract class of second-order evolution equations. It applies to the Berger plate with small in-plane stresses (and, depending on some open questions, possibly to the von Kármán model). This method partly relies on uniform stabilization and offers no apparent leeway to improve the controllability time. Strategy (ii) is demonstrated on the example of a Kirchhoff model with a dissipative polynomial source term. Such a source serves as a prototype for ultimately considering the same control construction for the Berger and von Kármán equations. The bound on the control time in this case is still open, but this method offers a conjectural possibility to sharpen the minimal control time estimate.

This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, utt − Δpu − Δut = f(u), in a bounded domain Ω ⊂ ℝ3 and subject to Dirichlét boundary conditions. The operator Δp, 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f(u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W01,p(Ω) into L2(Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

We study the well-posedness of a total linearization, with respect to a perturbation of the external forcing, of a free-boundary nonlinear elasticity--incompressible fluid interaction. The total linearization for the coupling modeled by the Navier--Stokes equations and the nonlinear equations of elastodynamics was obtained recently in [L. Bociu and J.-P. Zolésio, Evol. Equ. Control Theory, 2 (2013), pp. 55--79]. The equations and the free boundary were linearized together, and the result turned out to be quite different from the usual coupling of classical linear models. New additional terms are present on the common interface, some of them involving boundary curvatures and boundary acceleration. These terms play an important role in the final linearized system and cannot be neglected; their presence also introduces new challenges in the well-posedness analysis, which proceeds to establish that the evolution operator associated to the linearized system can be represented as a bounded perturbation of a maximal dissipative semigroup generator.

We establish existence of an optimal control for the problem of minimizing flow turbulence in the case of a nonlinear fluid-structure interaction model in the framework of the known local well-posedness theory. If the initial configuration is regular, in an appropriate sense, then a class of sufficiently smooth control inputs contains an element that minimizes, within the control class, the vorticity of the fluid flow around a moving and deforming elastic solid.

Presented here is a study of well-posedness and asymptotic stability of a "degenerately damped" PDE modeling a vibrating elastic string. The coefficient of the damping may vanish at small amplitudes thus weakening the effect of the dissipation. It is shown that the resulting dynamical system has strictly monotonically decreasing energy and uniformly decaying lower-order norms, however, is not uniformly stable on the associated finite-energy space. These theoretical findings were motivated by numerical simulations of this model using a finite element scheme and successive approximations. A description of the numerical approach and sample plots of energy decay are supplied. In addition, for certain initial data the solution can be determined in closed form up to a dissipative nonlinear ordinary differential equation. Such solutions can be used to assess the accuracy of the numerical examples.

This note gives a concise summary of results concerning the well-posedness and long-time behavior of (Reissner)–Mindlin–Timoshenko plate equations as presented in Pei et al. (Local and global well-posedness for semilinear Reissner–Mindlin–Timoshenko plate equations, 2013 and Global well-posedness and stability of semilinear Mindlin–Timoshenko system, 2013). The main feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms. The results include Hadamard local well-posedness, global existence, blow-up theorems, as well as estimates on the uniform energy decay rates.

We present a unified approach that bridges and extends a number of earlier results on stabilization of 2nd-order hyperbolic equations on manifolds. The methodology captures geometric requirements for damping acting simultaneously on subsets of the interior and of the boundary, and shows how placements of these feedbacks can complement each other depending on the underlying surface. In addition, the results conveniently incorporate the existing theory that allows elimination of geometric conditions from the controlled boundary (in absence of nearby interior damping), and elimination of damping entirely from certain boundary neighborhoods.

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

This is a study of local and global well-posedness of nonlinearly perturbed Reissner–Mindlin–Timoshenko plate equations. This PDE system represents an extension of the Timoshenko beam model to plates and accounts for shear deformations. The primary feature of the considered model is the interplay between nonlinear viscous interior damping and nonlinear source terms. The main results verify local and global existence of solutions as well as their continuous dependence on the initial data in the appropriate function spaces. Moreover, a blow-up result is proved for solutions with negative initial energy.

This paper settles a conjecture by Gazzola and Pavani [10] regarding solutions to the fourth order ODE w ( 4 ) + k w ″ + f ( w ) = 0 which arises in models of traveling waves in suspension bridges when k > 0 . Under suitable assumptions on the nonlinearity f and initial data, we demonstrate blow-up in finite time. The case k ≤ 0 was first investigated by Gazzola et al., and it is also handled here with a proof that requires less differentiability on f . Our approach is inspired by Gazzola et al. and exhibits the oscillatory mechanism underlying the finite-time blow-up. This blow-up is nonmonotone, with solutions oscillating to higher amplitudes over shrinking time intervals. In the context of bridge dynamics this phenomenon appears to be a consequence of mutually-amplifying interactions between vertical displacements and torsional oscillations.

We study long-term behavior of Reissner–Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. This work complements [28] which established local well-posedness of this problem, global well-posedness when damping dominates the sources (in an appropriate sense) and a blow-up in the complementary scenario assuming negative “total” initial energy. The current paper develops the potential well theory for the RMT system: it proves global existence for potential well solutions without restricting the source exponents, derives explicit energy decay rates dependent on the order of the damping exponents, and verifies a blow-up result for positive total initial energy.