Coupled parabolic–hyperbolic Stokes–Lamé PDE system: limit behaviour of the resolvent operator on the imaginary axis

This article is focused on an established, genuinely physical fluid-structure interaction model, whereby the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic–hyperbolic system of two partial differential equations in three dimensions with non-standard coupling at the boundary interface: the (dynamic) Stokes system (parabolic, modelling the fluid) and the Lamé system (hyperbolic, modelling the structure). This system generates a contraction semigroup on the natural energy space [G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: explicit semigroup generator and its spectral properties, Fluids and Waves, Amer. Math. Soc. Contemp. Math. 440 (2007), pp. 15–59] (canonical model) and [G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Series S, 2(3) (2009), pp. 417–447]. The boundary interface may or may not include a ‘damping’ (or dissipative) term. If damping is active on the entire interface, then uniform (exponential) stabilization is ensured, regardless of the geometry of the structure [G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst. 22(4) 2008, pp. 817–835, special issue, invited paper] (canonical model) and [G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system, J. Evol. Eqns 9(2009), pp. 341–370]. This article emphasizes the case of, at most, partial damping. At any rate, the main result is a precise uniform-operator limit behaviour of the resolvent operator of the semigroup generator on the imaginary axis of interest in itself, which holds true with or without damping. It, in turn, then implies a fortiori strong stability results: most notably, on the whole state space, under at least partial damping at the interface; and, in the absence of damping, on the whole state space, after factoring out an explicit one-dimensional null eigenspace, at least for a large class of geometries of the structure: these are characterized by a uniqueness property of a special over-determined elliptic problem.     

Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

We consider a parabolic–hyperbolic coupled system of two partial differential equations (PDEs), which governs fluid–structure interactions, and which features a suitable boundary dissipation term at the interface between the two media. The coupled system consists of Stokes flow coupled to the Lamé system of dynamic elasticity, with the respective dynamics being coupled on a boundary interface, where dissipation is introduced. Such a system is semigroup well-posed on the natural finite energy space (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). Here we prove that, moreover, such semigroup is uniformly (exponentially) stable in the corresponding operator norm, with no geometrical conditions imposed on the boundary interface. This result complements the strong stability properties of the undamped case (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). R. Triggiani’s research was partially supported by National Science Foundation under grant DMS-0104305 and by the Army Research Office under grant DAAD19-02-1-0179.     

The critical case of clamped thermoelastic systems with interior point control: Optimal interior and boundary regularity results

In the case of clamped thermoelastic systems with interior point control defined on a bounded domain Ω, the critical case is n=dimΩ=2. Indeed, an optimal interior regularity theory was obtained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004] for n=1 and n=3. However, in this reference, an ‘?-loss’ of interior regularity has occurred due to a peculiar pathology: the incompatibility of the B.C. of the spaces and . The present paper manages to establish that, indeed, one can take ?=0, thus obtaining an optimal interior regularity theory also for the case n=2. The elastic variables have the same interior regularity as in the corresponding elastic problem [R. Triggiani, Regularity with interior point control, Part II: Kirchhoff equations, J. Differential Equations 103 (1993) 394-421] (Kirchhoff). Unlike [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], the present paper establishes the sought-after interior regularity of the thermoelastic problem through a technical analysis based on sharp boundary (trace) regularity theory of Kirchhoff and wave equations. In the process, a new boundary regularity result, not contained in [R. Triggiani, Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case, Discrete Contin. Dyn. Syst. (Suppl.) (2007) 993-1004], is obtained for the elastic displacement of the thermoelastic system.     

Null controllability of von Kármán thermoelastic plates under the clamped or free mechanical boundary conditions

In this paper, we provide results of local and global null controllability for 2-D thermoelastic systems, in the absence of rotational inertia, and under the influence of the (nonLipschitz) von Kármán nonlinearity. The plate component may be taken to satisfy either the clamped or higher order (and physically relevant) free boundary conditions. In the accompanying analysis, critical use is made of sharp observability estimates which obtain for the linearization of the thermoelastic plate (these being derived in [G. Avalos, I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl. 294 (2004) 34-61] and [G. Avalos, I. Lasiecka, Asymptotic rates of blowup for the minimal energy function for the null controllability of thermoelastic plates: The free case, in: Proc. of the Conference for the Control of Partial Differential Equations, Georgetown University, Dekker, in press]). Moreover, another key ingredient in our work to steer the given nonlinear dynamics is the recent result in [A. Favini, M.A. Horn, I. Lasiecka, D. Tataru, Addendum to the paper: Global existence, uniqueness and regularity of solution to a von Kármán system with nonlinear boundary dissipation, Differential Integral Equations 10 (1997) 197-200] concerning the sharp regularity of the von Kármán nonlinearity.     

Interface feedback control stabilization of a nonlinear fluid-structure interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈Rⁿ, n=2, where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of the Navier-Stokes equation coupled on the boundary with the dynamic system of elasticity. We shall consider models where the elastic body exhibits small but rapid oscillations. These are established models arising in engineering applications when the structure is immersed in a viscous flow of liquid. Questions related to the stability of finite energy solutions are of paramount interest.It was shown in Lasiecka and Lu (2011) [14] that all data of finite energy produce solutions whose energy converges strongly to zero. The cited result holds under “partial flatness” geometric condition whose role is to control the effects of the pressure in the NS equation. Related conditions has been used in Avalos and Triggiani (2008) [23] for the analysis of the linear model. The goal of the present work is to study uniform stability of all finite energy solutions corresponding to nonlinear interaction. This particular question, of interest in its own rights, is also a necessary preliminary step for the analysis of optimal control strategies arising in infinite-horizon control problems associated with the structure. It is shown in this paper that a stress type feedback control applied on the interface of the structure produces solutions whose energy is exponentially stable.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.     

Strong solutions to a nonlinear fluid structure interaction system

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969] and considered in [V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55-82; V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173-1207]. In particular, the strong solutions here are obtained given initial datum for the Navier-Stokes equation in the space H¹, and initial data for the wave equation w₀ and w₁ in the spaces H²(Ωe) and H¹(Ωe) respectively.     

The null controllability of thermoelastic plates and singularity of the associated minimal energy function

We consider the null controllability problem for thermoelastic plates, defined on a two dimensional domain Ω, and subject to hinged, clamped or free boundary conditions. The uncontrolled partial differential equation system generates an analytic semigroup on the space of finite energy. Consequently, the concept of null controllability is indeed appropriate for consideration here. It is shown that all finite energy states can be driven to zero by means of just one L2((0,T)×Ω) control be it either mechanical or thermal. The singularity, as T↓0, of the associated minimal energy function is the main object studied in the paper. Singularity and blow-up rates for minimal energy function are not only of interest in their own right but are also of critical importance in Stochastic PDEs. In this paper, we establish the optimal blow-up rate for this function. It is shown that the rate of singularity is the same as for finite-dimensional truncations of the model. In view of sharp estimates available in the finite dimensional setting [Math. Control Signals Systems 9 (1997) 327], the singularity rates provided in this paper are optimal.     

Exact Controllability of a Semilinear Thermoelastic System with Control Solely in Thermal Equation

A result concerning the exact controllability of a semilinear thermoelastic system, in which the control term occurs solely in the thermal equation, is derived under the influence of rotational inertia and Lipschitz nonlinearity, subject to the clamped/Dirichlet boundary conditions. In the proof, we make use of the result given by Avalos (Differential and Integral Equations, 2000; 13(4–6):613–630), which states that the corresponding linear system is exact controllable.     

Exact controllability of a 3D piezoelectric body

In this Note we study the exact controllability of a three-dimensional body made of a material whose constitutive law introduces an elasticity-electricity coupling. We show that, without any geometrical assumption, two controls (the elastic and the electric controls) acting on the whole boundary drive the system to rest in finite time. To cite this article: I. Lasiecka, B. Miara, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Min-max game theory and nonstandard differential Riccati equations for abstract hyperbolic-like equations

We consider the abstract dynamical framework of Lasiecka and Triggiani (2000) [1, Chapter 9], which models a large variety of mixed PDE problems (see specific classes in the Introduction) with boundary or point control, all defined on a smooth, bounded domain Ω⊂Rⁿ, n arbitrary. This means that the input → solution map is bounded on natural function spaces. We then study min-max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.     

Symbolic dynamics: from the N-centre to the (N + 1)-body problem,a preliminary study

We consider a rotating N-centre problem, with N ≥ 3 and homogeneous potentials of degree ?α < 0, α ∈ [1,2). We prove the existence of infinitely many collision-free periodic solutions with negative and small Jacobi constant and small values of the angular velocity, for any initial configuration of the centres. We will introduce a Maupertuis’ type variational principle in order to apply the broken geodesics technique developed in Soave and Terracini (Discrete Contin Dyn Syst 32:3245–3301, 2012). Major difficulties arise from the fact that, contrary to the classical Jacobi length, the related functional does not come from a Riemaniann structure but from a Finslerian one. Our existence result allows us to characterize the associated dynamical system with a symbolic dynamics, where the symbols are given partitions of the centres in two non-empty sets.     

Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers

The present paper seeks to continue the analysis in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on tangential boundary stabilization of Navier–Stokes equations, d=2,3$d=2,3$, as deduced from well-posedness and stability properties of the corresponding linearized equations. It intends to complement [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on two levels: (i) by casting the Riccati-based results of Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] for d=2,3$d=2,3$ in an abstract setting, thus extracting the key relevant features, so that the resulting framework may be applicable also to other stabilizing boundary feedback operators, as well as to other parabolic-like equations of fluid dynamics; (ii) by including, in the case d=2$d=2$ this time, also the low-level gain counterpart of the results in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] with both Riccati-based and spectral-based (tangential) feedback controllers. This way, new local boundary stabilization results of Navier–Stokes equations are obtained over [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear].     

On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation with f(0)=a, f^′(0)=1, f^′(∞):=limt→∞f^′(t)=0, where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β?−2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929-948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311-351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67-73], we conclude that the given problem may possess at most two types solutions for β∈R. Moreover, multiple solutions are also verified for various pairs of (a,β).     

L∞ bounds for solutions of parametrized elliptic equations

This generalizes earlier results (T. I. Seidman, Indiana Univ. Math. J.30 (1981), 305–311) for ?Δu = λf(u). For the family of equations (su1) Au = g(u, λ) with appropriate boundary conditions the object is to construct from g and the boundary conditions a function η(λ, r) such that a bound y(λ) on ∥u∥_∞ can be obtained by solving the ODE: y′(λ) = η(λ, y) with y(λ₀) = B(λ₀) = bound at λ = λ₀.     

Exact controllability of a semilinear thermoelastic system with control and nonlinearity in thermal component only

A result concerning the exact controllability of semilinear thermoelastic system, in which the control and nonlinear term occurs solely in the thermal equation, is derived under the influence of rotational inertia and Lipschitz nonlinearity, subject to the clamped/Dirichlet boundary conditions. In the proof we make use the result given by Avalos G. [Differential and Integral Equations, 13 (2000), 613-630] which states that the corresponding linear system is exact controllable. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Products of threads

A topological semigroup S is a thread if S is a metric arc in which one endpoint is the identity of S and the other endpoint is the zero of S. Let [x0, x₁] and [y₀, y₁] be subsets of threads in a semigroup S. Then define A(x₀, x₁) = x [x₀, x₁] | xy₁ = x₁y for some y [y₀, y[in1}]). The main result of this paper states that if X and Y are threads in a topological semigroup S, then XY is an arc or a point, or contains a two-cell.This paper is part of the author's doctoral dissertation written under the direction of Professor D. R. Brown at the University of Houston. This work was supported in part by an NSF Science Faculty Fellowship.     

Existence and Exponential Decay of Solutions to a Quasilinear Thermoelastic Plate System

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in , n ≤ 3. Existence of finite energy solutions describing the dynamics of a nonlinear thermoelastic plate is established. In addition asymptotic long time behavior of weak solutions is discussed. It is shown that finite energy solutions decay exponentially to zero with the rate depending only on the (finite energy) size of initial conditions. The proofs are based on methods of weak compactness along with nonlocal partial differential operator multipliers which supply the sought after “recovery” inequalities. Regularity of solutions is also discussed by exploiting the underlying analyticity of the linearized semigroup along with a related maximal parabolic regularity [1, 16, 44]. The research of I. Lasiecka has been partially supported by DMS-NSF Grant Nr 0606882. S. Maad was supported by the Swedish Research Council and by the European Union under the Marie Curie Fellowship MEIF-CT-2005-024191.     

Constructive existence and uniqueness for some nonlinear two-point boundary value problems

Constructive existence and uniqueness theorems are presented for the problem $\text{y″ = ?(x, y), y(0) = y}{}_0\text{, y(1) = y}{}_1$. Applications to several problems are also given including one in which the boundary values are y′(0) = y′₀, y(1) = y₁.     

On the Möbius function of the locally finite poset associated with a numerical semigroup

Let S be a numerical semigroup and let (?,≤ _S ) be the (locally finite) poset induced by S on the set of integers ? defined by x≤ _S y if and only if y?x∈S for all integers x and y. In this paper, we investigate the Möbius function associated to (?,≤ _S ) when S is an arithmetic semigroup.