Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions
Barbara Kaltenbacher Irena Lasiecka
In this paper we show wellposedness of two equations of nonlinear acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider Neumann boundary conditions which are of particular practical interest in applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations with potential degeneration and strong damping. We prove local in time well-posedness as well as global existence and exponential decay for a slightly modi ed model. A key step of the proof is an appropriate extension of the Neumann boundary data to the interior along with exploitation of singular estimates associated with the analytic semigroup generated by the strongly damped wave equation.
keywords: nonlinear acoustics local and global well- posedness Kuznetsov's equation
Eliminating flutter for clamped von Karman plates immersed in subsonic flows
Irena Lasiecka Justin Webster
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.
keywords: mathematical aeroelasticity flutter nonlinear plates Strong stability nonlinear semigroups.
Alain Haraux Irena Lasiecka
The present Inaugural Volume is the first Issue of a new journal Evolution Equations and Control Theory [EECT], which is published within the AIMS Series. EECT is devoted to topics lying at the interface between Evolution Equations and Control Theory of Dynamics. Evolution equations are to be understood in a broad sense as Infinite Dimensional Dynamics which often arise in modeling physical systems as an infinite-dimensional process. This includes single PDE (Partial Differential Equations) or FDE (Functional Differential Equations) as well as coupled dynamics of different characteristics with an interface between them. Since modern control theory intrinsically depends on a good understanding of the qualitative theory of dynamics and evolution theory, the choice of these two topics appears synergistic and most natural. Past experience shows that new developments in control theory often depend on sufficient information related to the associated dynamical properties of the system. On the other hand, developments in evolution theory allow one to consider certain control theoretic formulations that alone would not appear treatable.

For more information please click the "Full Text" above.
Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects
Marcelo M. Cavalcanti Valéria N. Domingos Cavalcanti Irena Lasiecka Flávio A. Falcão Nascimento
Wave equation defined on a compact Riemannian manifold $(M, \mathfrak{g})$ subject to a combination of locally distributed viscoelastic and frictional dissipations is discussed. The viscoelastic dissipation is active on the support of $a(x)$ while the frictional damping affects the portion of the manifold quantified by the support of $b(x)$ where both $a(x)$ and $b(x)$ are smooth functions. Assuming that $a(x) + b(x) \geq \delta >0 $ for all $x\in M$ and that the relaxation function satisfies certain nonlinear differential inequality, it is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping. In the special case when the viscoelastic effect is active on the entire domain and the frictional dissipation is differentiable at the origin, then the overall decay rates are dictated by the viscoelasticity. The obtained decay estimates are intrinsic without any prior quantification of decay rates of both viscoelastic and frictional dissipative effects. This particular topic has been motivated by influential paper of Fabrizio-Polidoro [15] where it was shown that viscoelasticity with poorly behaving relaxation kernel destroys exponential decay rates generated by linear frictional dissipation. In this paper we extend these considerations to: (i) nonlinear dissipation with unquantified growth at the origin (frictional) and infinity (viscoelastic) , (ii) more general geometric settings that accommodate competing nature of frictional and viscoelastic damping.
keywords: viscoelastic and frictional distributed damping compact Riemanian manifold decay rates. Wave equation
Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions
Nicolas Fourrier Irena Lasiecka
We present an analysis of regularity and stability of solutions corresponding to wave equation with dynamic boundary conditions. It has been known since the pioneering work by [26, 27, 30] that addition of dynamics to the boundary may change drastically both regularity and stability properties of the underlying system. We shall investigate these properties in the context of wave equation with the damping affecting either the interior dynamics or the boundary dynamics or both.
    This leads to a consideration of a wave equation acting on a bounded 3-d domain coupled with another second order dynamics acting on the boundary. The wave equation is equipped with a viscoelastic damping, zero Dirichlet boundary conditions on a portion of the boundary and dynamic boundary conditions. These are general Wentzell type of boundary conditions which describe wave equation oscillating on a tangent manifold of a lower dimension. We shall examine regularity and stability properties of the resulting system -as a function of strength and location of the dissipation. Properties such as well-posedness of finite energy solutions, analyticity of the associated semigroup, strong and uniform stability will be discussed.
    The results obtained analytically are illustrated by numerical analysis. The latter shows the impact of various types of dissipation on the spectrum of the generator as well as the dynamic behavior of the solution on a rectangular domain.
keywords: Wave equation with dynamic boundary conditions Laplace'a Beltrami operator Gevrey's regularity strong stability spectral analysis. uniform stability strong damping analyticity of semigroups semigroup generation
Regularity of boundary traces for a fluid-solid interaction model
Francesca Bucci Irena Lasiecka
We consider a mathematical model for the interactions of an elastic body fully immersed in a viscous, incompressible fluid. The corresponding composite PDE system comprises a linearized Navier-Stokes system and a dynamic system of elasticity; the coupling takes place on the interface between the two regions occupied by the fluid and the solid, respectively. We specifically study the regularity of boundary traces (on the interface) for the fluid velocity field. The obtained trace regularity theory for the fluid component of the system-of interest in its own right-establishes, in addition, solvability of the associated optimal (quadratic) control problems on a finite time interval, along with well-posedness of the corresponding operator Differential Riccati equations. These results complement the recent advances in the PDE analysis and control of the Stokes-Lamé system.
keywords: trace regularity optimal control problems. Fluid-solid interactions
Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers
Irena Lasiecka Roberto Triggiani
We consider a heat--structure interaction model where the structure is subject to viscoelastic (strong) damping. This is a preliminary step toward the study of a fluid--structure interaction model where the heat equation is replaced by the linear version of the Navier--Stokes equation as it arises in applications. We prove four main results: analyticity of the corresponding contraction semigroup (which cannot follow by perturbation); sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point $\lambda=-1$ in its continuous spectrum; exponential decay of the semigroup with sharp decay rate; finally, a characterization of the domains of fractional power related to the generator.
keywords: fractional powers. analyticity viscoelastic damping Heat-structure
Global existence and exponential decay rates for the Westervelt equation
Barbara Kaltenbacher Irena Lasiecka
We consider the Westervelt equation which models propagation of sound in a fluid medium. This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging and therapy. The PDE model consists of the second order in time evolution which is both quasilinear and degenerate. Degeneracy depends on the fluctuations of the acoustic pressure.
    Our main results are : (1) global well-posedness, (2) exponential decay rates for the energy function corresponding to both weak and strong solutions. The proof is based on (i) application of a suitable fixed point theorem applied to an appropriate formulation of the PDE which exhibits analyticity properties of the underlying linearised semigroup, (ii) exploitation of decay rates associated with the dissipative mechanism along with barrier's method leading to global wellposedness. The obtained result holds for all times, provided that the initial data are taken from a suitably small ball characterized by the parameters of the equation.
keywords: Westervelt equation exponential decay. global existence
Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system
Irena Lasiecka Mathias Wilke
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
keywords: existence and uniqueness of strong solutions maximal regularity Quasilinear thermoelastic plates exponential decay.
Asymptotic behavior of solutions in nonlinear dynamic elasticity
Irena Lasiecka W. Heyman
A Dynamic system of 2-D nonlinear elasticity with nonlinear interior dissipation is considered. It is assumed that the principal part of elastic operator is perturbed by the unstructured lower order linear terms. Asymptotic behavior of solutions when time $t \rightarrow 0$ is analyzed. It is shown that in the case of zero load applied to the plate, the arbitrarily large decay rates can be achieved provided that both the "damping" coefficient and the "traction" coefficient are suitably large. This result generalizes and extends, to the nonlinear and multidimensional context, the earlier results obtained only for the one-dimensional linear wave equation. In the case of a loaded plate the existence of compact global attractor attracting all solutions is established.
keywords: asymptotic behavior nonlinear elasticity nonlinear interior dissipation decay rates.

Year of publication

Related Authors

Related Keywords

[Back to Top]