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Irena Lasiecka

Publications and source records attributed to Irena Lasiecka.

At least 37 records · Page 2Linked to original sources
arXiv2018

Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound

We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev $H^{k}$-spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small in a lower topology (basic energy level corresponding to weak solutions), we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.

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arXiv2018

Exponential Decay of Quasilinear Maxwell Equations with Interior Conductivity

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of $\mathbb{R}^{3}$ with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical $L^{2}$-Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time $H^{3}$-solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy and the lack of solenoidality, etc., are addressed. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate.

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arXiv2018

Feedback control of the acoustic pressure in ultrasonic wave propagation

Classical models for the propagation of ultrasound waves are the Westervelt equation, the Kuznetsov and the Khokhlov-Zabolotskaya-Kuznetsov equations. The Jordan-Moore-Gibson-Thompson equation is a prominent example of a Partial Differential Equation (PDE) model which describes the acoustic velocity potential in ultrasound wave propagation, where the paradox of infinite speed of propagation of thermal signals is eliminated; the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law, accounts for its being of third order in time. Aiming at the understanding of the fully quasilinear PDE, a great deal of attention has been recently devoted to its linearization -- referred to in the literature as the Moore-Gibson-Thompson equation -- whose mathematical analysis is also of independent interest, posing already several questions and challenges. In this work we consider and solve a quadratic control problem associated with the linear equation, formulated consistently with the goal of keeping the acoustic pressure close to a reference pressure during ultrasound excitation, as required in medical and industrial applications. While optimal control problems with smooth controls have been considered in the recent literature, we aim at relying on controls which are just $L^2$ in time; this leads to a singular control problem and to non-standard Riccati equations. In spite of the unfavourable combination of the semigroup describing the free dynamics that is not analytic, with the challenging pattern displayed by the dynamics subject to boundary control, a feedback synthesis of the optimal control as well as well-posedness of operator Riccati equations are established.

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arXiv2016

Global Existence and Exponential Stability for a Nonlinear Thermoelastic Kirchhoff-Love Plate

We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff \& Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.

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arXiv2016

Quasi-stability and Exponential Attractors for A Non-Gradient System---Applications to Piston-Theoretic Plates with Internal Damping

We consider a nonlinear (Berger or Von Karman) clamped plate model with a {\em piston-theoretic} right hand side---which include 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 depending 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.

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arXiv2015

Mathematical Aeroelasticity: A Survey

A variety of models describing the interaction between flows and oscillating structures are discussed. The main aim is to analyze conditions under which structural instability (flutter) induced by a fluid flow can be suppressed or eliminated. The analysis provided focuses on effects brought about by: (i) different plate and fluid boundary conditions, (ii) various regimes for flow velocities: subsonic, transonic, or supersonic, (iii) different modeling of the structure which may or may not account for in-plane accelerations (full von Karman system), (iv) viscous effects, (v) an assortment of models related to piston-theoretic model reductions, and (iv) considerations of axial flows (in contrast to so called normal flows). The discussion below is based on conclusions reached via a combination of rigorous PDE analysis, numerical computations, and experimental trials.

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arXiv2015

Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow

Asymptotic-in-time feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [22] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of imposed energy dissipation the plate dynamics converge to a compact and finite dimensional set [16,17]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the stationary set in the natural energy topology. In doing so, we demonstrate the existence of an exponential attractor for the plate dynamics. That the exponential attractor exhibits additional smoothness is the technical crux of the main result. This property cannot be taken for granted, as exponential attractors are often compact but not necessarily smooth (in contrast with global maximal attractors). Our result implies that flutter (a periodic or chaotic end behavior) can be eliminated (in subsonic flows) with sufficient frictional damping in the structure. While such a result has been proved in the past for regularized plate models (with rotational inertia terms or thermal considerations [14,33,38,39]), this is the first treatment which does not incorporate smoothing effects for the structure.

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arXiv2015

Von Karman plate in a gas flow: recent results and conjectures

We give a survey of recent results on 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 considered. The focus of the discussion here is on the interesting mathematical aspects of physical phenomena occurring in aeroelasticity, such as flutter and divergence. This leads to a partial differential equation (PDE) treatment of issues such as well-posedness of finite energy solutions, and long-time (asymptotic) behavior. The latter includes theory of asymptotic stability, convergence to equilibria, and to global attracting sets. We complete the discussion with several well known observations and conjectures based on experimental/numerical studies.

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arXiv2015

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.

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arXiv2015

Moore-Gibson-Thompson equation with memory, part II: general decay of energy

We study a temporally third order (Moore-Gibson-Thompson) equation with a memory term. Previously it is known that, in non-critical regime, the global solutions exist and the energy functionals decay to zero. More precisely, it is known that the energy has exponential decay if the memory kernel decays exponentially. The current work is a generalization of the previous one (Part I) in that it allows the memory kernel to be more general and shows that the energy decays the same way as the memory kernel does, exponentially or not.

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arXiv2015

Moore-Gibson-Thompson equation with memory, part I: exponential decay of energy

We are interested in the Moore-Gibson-Thompson(MGT) equation with memory \begin{equation}\nonumber τu_{ttt}+ αu_{tt}+c^2\A u+b\A u_t -\int_0^tg(t-s)\A w(s)ds=0. \end{equation} We first classify the memory into three types. Then we study how a memory term creates damping mechanism and how the memory causes energy decay.

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arXiv2013

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.

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arXiv2013

Nonlinear Plates Interacting with A Subsonic, Inviscid Flow via Kutta-Joukowski Interface Conditions

We analyze the well-posedness of a flow-plate interaction considered in [22, 24]. Specifically, we consider the Kutta-Joukowski boundary conditions for the flow [20, 28, 26], which ultimately give rise to a hyperbolic equation in the half-space (for the flow) with mixed boundary conditions. This boundary condition has been considered previously in the lower-dimensional interactions [1, 2], and dramatically changes the properties of the flow-plate interaction and requisite analytical techniques. We present results on well-posedness of the fluid-structure interaction with the Kutta-Joukowsky flow conditions in force. The semigroup approach to the proof utilizes an abstract setup related to that in [16] but requires (1) the use of a Neumann-flow map to address a Zaremba type elliptic problem and (2) a trace regularity assumption on the acceleration potential of the flow. This assumption is linked to invertibility of singular integral operators which are analogous to the finite Hilbert transform in two dimensions. (We show the validity of this assumption when the model is reduced to a two dimensional flow interacting with a one dimensional structure; this requires microlocal techniques.) Our results link the analysis in [16] to that in [1, 2].

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arXiv2012

Attractors for Delayed, Non-Rotational von Karman Plates with Applications to Flow-Structure Interactions Without any Damping

This paper is devoted to a long time behavior analysis associated with flow structure interactions at subsonic and supersonic velocities. It turns out that an ntrinsic component of that analysis is the study of attracting sets corresponding to von Karman plate equations with {\it delayed terms} and {\it without rotational} terms. The presence of delay terms in the dynamical system leads to the loss of gradient structure while the absence of rotational terms in von Karman plates leads to the loss of compactness of the orbits. Both these features make the analysis of long time behavior rather subtle rendering the established tools in the theory of PDE dynamical systems not applicable. It is our goal to develop methodology that is capable of handling this class of problems.

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arXiv2012

Evolution Semigroups in Supersonic Flow-Plate Interactions

We consider the well-posedness of a model for a flow-structure interaction. This model describes the dynamics of an elastic flexible plate with clamped boundary conditions immersed in a supersonic flow. A perturbed wave equation describes the flow potential. The plate's out-of-plane displacement can be modeled by various nonlinear plate equations (including von Karman and Berger). We show that the linearized model is well-posed on the state space (as given by finite energy considerations) and generates a strongly continuous semigroup. We make use of these results to conclude global-in-time well-posedness for the fully nonlinear model. The proof of generation has two novel features, namely: (1) we introduce a new flow potential velocity-type variable which makes it possible to cover both subsonic and supersonic cases, and to split the dynamics generating operator into a skew-adjoint component and a perturbation acting outside of the state space. Performing semigroup analysis also requires a nontrivial approximation of the domain of the generator. And (2) we make critical use of hidden regularity for the flow component of the model (in the abstract setup for the semigroup problem) which allows us run a fixed point argument and eventually conclude well-posedness. This well-posedness result for supersonic flows (in the absence of rotational inertia) has been hereto open. The use of semigroup methods to obtain well-posedness opens this model to long-time behavior considerations.

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arXiv2012

Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents

These lectures present the analysis of stability and control of long time behavior of PDE models described by nonlinear evolutions of hyperbolic type. Specific examples of the models under consideration include: (i) nonlinear systems of dynamic elasticity: von Karman systems, Berger's equations, Kirchhoff - Boussinesq equations, nonlinear waves (ii) nonlinear flow - structure and fluid - structure interactions, (iii) and nonlinear thermo-elasticity. A characteristic feature of the models under consideration is criticality or super-criticality of sources (with respect to Sobolev's embeddings) along with super-criticality of damping mechanisms which, in addition, may be also geometrically constrained. Our aim is to present several methods relying on cancelations, harmonic analysis and geometric analysis, which enable to handle criticality and also super-criticality in both sources and the damping of the underlined nonlinear PDE. It turns out that if carefully analyzed the nonlinearity can be taken "advantage of" in order to produce implementable damping mechanism. Another goal of these lectures is the understanding of control mechanisms which are geometrically constrained. The final task boils down to showing that appropriately damped system is "quasi-stable" in the sense that any two trajectories approach each other exponentially fast up to a compact term which can grow in time. Showing this property- formulated as quasi-stability estimate -is the key and technically demanding issue that requires suitable tools. These include: weighted energy inequalities, compensated compactness, Carleman's estimates and some elements of microlocal analysis.

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arXiv2012

A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control

We study the infinite horizon Linear-Quadratic problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of Partial Differential Equations (PDE) with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDE with an overall `predominant' hyperbolic character, such as, e.g., some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics. In the present case, even the more general theory appealing to estimates of the singularity displayed by the kernel which occurs in the integral representation of the solution to the control system fails. A novel framework which embodies possible hyperbolic components of the dynamics has been introduced by the authors in 2005, and a full theory of the LQ-problem on a finite time horizon has been developed. The present paper provides the infinite time horizon theory, culminating in well-posedness of the corresponding (algebraic) Riccati equations. New technical challenges are encountered and new tools are needed, especially in order to pinpoint the differentiability of the optimal solution. The theory is illustrated by means of a boundary control problem arising in thermoelasticity.

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