Serge Nicaise

Polytechnic University of Hauts-de-France | UPHF · Laboratoire de Mathématiques et leurs Applications de Valenciennes (LAMAV)

We study the free Schrödinger equation on finite metric graphs with infinite ends. We give sufficient conditions to obtain the L 1 ( R ) → L ∞ ( R ) time decay rate at least t−1/2. These conditions allow certain metric graphs with circles and/or with commensurable lengths of the bounded edges. Further we study the dynamics of the probability flow b...

The purpose of this work is to investigate the stabilization of locally weakly coupled second‐order evolution equations of hyperbolic type, where only one of the two equations is directly damped. As such system cannot be exponentially stable, we are interested in polynomial energy decay rates. Our main contributions concern strong abstract and poly...

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected elastic–piezoelectric–elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected piezoelectric–elastic design with a local damping acting on the elastic par...

In this work, we propose an a posteriori goal-oriented error estimator for the harmonic $\textbf {A}$-$\varphi $ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a gu...

This study investigates the stability of a transmission problem featuring alternating magnetizable piezoelectric and elastic beams under various partial damping scenarios in five distinct cases.

In this paper, the stability of longitudinal vibrations for transmission problems of two smart-system designs are studied: (i) a serially-connected Elastic-Piezoelectric-Elastic design with a local damping acting only on the piezoelectric layer and (ii) a serially-connected Piezoelectric-Elastic design with a local damping acting on the elastic par...

In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate t-12\documentclass[12pt]{minimal} \usepa...

We consider Helmholtz problems in three-dimensional domains featuring conical points. We focus on the high-frequency regime and derive novel sharp upper-bounds for the stress intensity factors of the singularities associated with the conical points. We then employ these new estimates to analyse the stability of finite element discretisations. Our k...

A Correction to this paper has been published: 10.1007/s00028-021-00715-0

In this paper, we study the direct/indirect stability of locally coupled wave equations with local Kelvin-Voigt dampings/damping, where we assume that the supports of the dampings and the coupling coefficients are disjoint. First, we prove the well-posedness, strong stability, and polynomial stability for some one dimensional coupled systems. Moreo...

In this paper, we consider the Kirchhoff plate equation with delay terms on the dynamical boundary controls. We prove its well-posedness, strong stability, non-exponential stability, and polynomial stability under a multiplier geometric control condition.

We study the curl‐div‐system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal H‐convergence for closed Hilbert complexes, we define the appropriate topology for possibly nonlocal and non‐periodic coefficients in curl‐div systems to model highly oscillatory behavio...

In this paper, we study the direct/indirect stability of locally coupled wave equations with local Kelvin-Voigt dampings/damping and by assuming that the supports of the dampings and the coupling coefficients are disjoint. First, we prove the well-posedness, strong stability, and polynomial stability for some one dimensional coupled systems. Moreov...

We consider the well-posedness and the long time behavior of the Moore– Gibson–Thompson equation with memory in the critical case. We first find general sufficient conditions that guarantee a (optimal) polynomial decay of the system. Then by comparing the behavior of the resolvent of the Moore–Gibson–Thompson system with the one of the resolvent of...

In this paper, we investigate a network of elastic and thermo-elastic materials. On each thermo-elastic edge, we consider two coupled wave equations such that one of them is damped via a coupling with a heat equation. On each elastic edge (undamped), we consider two coupled conservative wave equations. Under some conditions, we prove that the therm...

In this work, we study the well-posedness and some stability properties of a PDE system that models the propagation of light in a metallic domain with a hole. This model takes into account the dispersive properties of the metal. It consists of a linear coupling between Maxwell's equations and a wave type system. We prove that the problem is well po...

In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $t^{-\frac{1}{2}}$ if the two waves have...

In this paper, we obtain some stability results of (abstract) dissipative evolution equations with a nonautonomous and nonlinear damping using the exponential stability of the retrograde problem with a linear and autonomous feedback and a comparison principle. We then illustrate our abstract statements for different concrete examples, where new res...

In this paper, we consider two models of the Kirchhoff plate equation, the first one with delay terms on the dynamical boundary controls (see system (1.1) below), and the second one where delay terms on the boundary control are added (see system (1.2) below). For the first system, we prove its well-posedness, strong stability, non-exponential stabi...

In this paper, we study the indirect stabilization of a coupled string-beam system related to the well known Lazer-McKenna suspension bridge model. We prove some decay results of the energy of the system with either interior dampings or boundary ones. Our method is based on observablility estimates of the undamped system and on the spectral analysi...

In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable....

In this paper, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin–Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt–Batty, we prove the strong stability of our system. Finally, using a frequency domai...

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in Kramar et al. (Linear hyperbolic systems on networks. arXiv:2003.08281, 2020) by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type—or a combination of...

The quality of a local physical quantity obtained by the numerical method such as the finite element method (FEM) attracts more and more attention in computational electromagnetism. Inspired by the idea of goal-oriented error estimate given for the Laplace problem, this work is devoted to a guaranteed a posteriori error estimate adapted for the qua...

This work deals with the finite element approximation of a prestressed shell model using a new formulation where the unknowns (the displacement and the rotation of fibers normal to the midsurface) are described in Cartesian and local covariant basis respectively. Due to the constraint involved in the definition of the functional space, a penalized...

We consider the wave equation with a delay term in the dynamical control. If the delay term is small enough, we establish the strong stability of this system, but show that it is not exponentially stable. We then prove that the system with delay has the same rational decay rate than the one without delay.

In this paper, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients. First, using a general criteria of Arendt–Batty, we prove the strong stability of our system. Second, using a frequency domain approach combined with the mu...

In this paper, we obtain some stability results of systems corresponding to the coupling between a dissipative evolution equation (set in an infinite dimensional space) and an ordinary differential equation. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro...

We study hyperbolic systems of one - dimensional partial differential equations under general , possibly non-local boundary conditions. A large class of evolution equations, either on individual 1- dimensional intervals or on general networks , can be reformulated in our rather flexible formalism , which generalizes the classical technique of first...

In this paper, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin-Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Finally, using a frequency domai...

The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general cri...

The A − φ − B magnetodynamic Maxwell system given in its potential and space-time formulation is a popular model considered in the engineering community. It allows to model some phenomena such as eddy current losses in multiple turn winding. Indeed, in some cases, they can significantly alter the performance of the devices, and consequently can no...

In this paper, we investigate the stabilization of a locally coupled wave equations with local viscoelastic damping of past history type acting only in one equation via non smooth coefficients. First, using a general criteria of Arendt-Batty, we prove the strong stability of our system. Second, using a frequency domain approach combined with the mu...

We consider a nonconforming hp‐finite element approximation of a variational formulation of the time‐harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H¹ as soon as the boundary is smooth enough (in particular it holds for domains...

In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable....

We study evolution equations on networks that can be modeled by means of hyperbolic systems. We extend our previous findings in \cite{KraMugNic20} by discussing well-posedness under rather general transmission conditions that might be either of stationary or dynamic type - or a combination of both. Our results rely upon semigroup theory and element...

We consider abstract second order evolution equations with unbounded feedback with delay. If the delay term is small enough, we rigorously prove the fact that the system with delay has the same decay rate than the one without delay. Some old and new results easily follow.

The purpose of this work is to investigate the stabilization of a system of weakly coupled wave equations with one or two locally internal Kelvin–Voigt damping and non-smooth coefficient at the interface. The main novelty in this paper is that the considered system is a coupled system and that the geometrical situations covered (see Remarks 5.6, 5....

The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus a finite number of isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general \(r_i^{\pi /(2\alpha _...

We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. A passitivity assumption guarantees the boundedness of the associated semigroup. Further the exponential or polynomial decay of the energy is proved under suit...

We analyze the stability of a linearized hydrodynamical model describing the response of nanometric dispersive metallic materials illuminated by optical light waves that is the situation occurring in nanoplasmonics. This model corresponds to the coupling between the Maxwell system and a PDE describing the evolution of the polarization current of th...

We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks, can be reformulated in our rather flexible formalism, which generalizes the classical technique of first-order...

Consider the Poisson equation in a polyhedral domain with mixed boundary conditions. We establish new regularity results for the solution with possible vertex and edge singularities with interior data in usual Sobolev spaces {H^{\sigma}} with {\sigma\in[0,1)} . We propose anisotropic finite element algorithms approximating the singular solution in...

This work deals with the finite element approximation of a prestressed shell model formulated in Cartesian coordinates system. The considered constrained variational problem is not necessarily positive. Moreover, because of the constraint, it cannot be discretized by conforming finite element methods. A penalized version of the model and its discre...

We derive stability estimates in H2 for elliptic problems with impedance boundary conditions that are uniform with respect to the impedance coefficient. Such estimates are of importance to establish sharp error estimates for finite element discretizations of contact impedance and high-frequency Helmholtz problems. Though stability in H2 is easily o...

We analyze the convergence of finite element discretizations of time-harmonic wave propagation problems. We propose a general methodology to derive stability conditions and error estimates that are explicit with respect to the wavenumber. This methodology is formally based on an expansion of the solution in powers of k, which permits to split the s...

The hinged Kirchhoff plate model contains a fourth order elliptic differential equation complemented with a zeroeth and a second order boundary condition. On domains with boundaries having corners the strong setting is not well‐defined. We here allow boundaries consisting of piecewise C2, 1‐curves connecting at corners. For such domains different v...

The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general $r_i^{\pi/(2\alpha_i)}$ at transition poi...

Transmission problems with sign changing coefficients and polyhedral interfaces are investigated. We discuss how such problems may enter a general elliptic theory with corners (standard model). Slides of an invited presentation.

We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. A passitivity assumption guarantees the boundedness of the associated semigroup. Further the exponential or polynomial decay of the energy is proved under suit...

In this paper, we first develop a variational formulation of the time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains similar to the one for domains with smooth boundary proposed in Section 4.5.d of Costabel et al., Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth Domains,...

We study the curl-div-system with variable coefficients and a nonlocal homogenisation problem associated with it. Using, in part refining, techniques from nonlocal $H$-convergence for closed Hilbert complexes, we define the appropriate topology for possibly nonlocal and non-periodic coefficients in curl-div systems to model highly oscillatory behav...

Motivated by different models of damped nanorods and nanoplates, we introduce an abstract framework of non local damped second order evolution equations. We analyze the stability and the asymptotic behavior of such evolution equations. We then illustrate our general results by some stability results for some models of damped nanorods and nanoplates...

This paper is devoted to the leader–follower consensus problem for high-order multi-agent systems with inherent nonlinear dynamics evolving on an arbitrary time domain. This problem is investigated using the time scale theory. Based on this theory, some sufficient conditions are derived to guarantee that the tracking errors exponentially converge t...

The problem of distributed leader‐follower consensus for second‐order linear multiagent systems with unknown nonlinear inherent dynamics is investigated in this paper. It is assumed that the dynamic of each agent is described by a semilinear second‐order dynamic equation on an arbitrary time scale. Using calculus on time scales and direct Lyapunov'...

In this paper a guaranteed equilibrated error estimator is developed for the 3D harmonic magnetodynamic problem of Maxwell's system. This system is recasted in the classical A−φ potential formulation and solved by the Finite Element method. The error estimator is built starting from the A−φ numerical solution by a local flux reconstruction techniqu...

We consider abstract quasilinear evolution equations of Sobolev type in a Hilbert setting. We propose two fully discrete schemes and prove some error estimates under minimal assumptions. Various examples that enter into our abstract framework are considered, for each of them our theoretical results are confirmed by several numerical experiments.

We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the "amplitude" of the singular parts is bounded explicitly in terms of frequency. We show that for h...

We analyze the stability of a dispersive medium immersed in vacuum (with Silver-Müller boundary condition in the exterior boundary) or vice versa. The dispersive medium model corresponds to the coupling between Maxwell's system and a first order ordinary differential equation (of parabolic type). For a dispersive medium coupled with vacuum, the ord...

In this paper, we deal with some magnetostatic models considered in vector potential formulations and solved by a Finite Element solver. In order to ensure the uniqueness of the solution, a gauge condition has to be imposed, and several possibilities occur. Moreover, the source term has to be correctly defined to ensure a physically admissible solu...

We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the $C_0$-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a gene...

We present a fast Galerkin spectral method to solve logarithmic singular equations on segments. The proposed method uses weighted first-kind Chebyshev polynomials. Convergence rates of several orders are obtained for fractional Sobolev spaces $H^{-1/2}$ (or $H^{-1/2}_{00}$). Main tools are the approximation properties of the discretization basis, t...

Consider the Poisson equation on a polyhedral domain with the given data in a weighted \(L^2\) space. We establish new regularity results for the solution with possible vertex and edge singularities and propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical met...

The first aim of this paper is to give different necessary and sufficient conditions that guarantee the density of the set of compactly supported functions into the Sobolev space of order one in infinite p-adic weighted trees. The second goal is to define properly a trace operator in this Sobolev space if it makes sense.

We construct classical solutions to the nonlinear Maxwell system with periodic boundary conditions which blow up in H(curl). A similar result is shown on the full space. Our construction is based on an analysis of a shock wave in one space dimension.

In this paper, we first introduce an abstract viscous hyperbolic problem for which we prove exponential decay under appropriated assumptions. We then give some illustrative examples, like the linearized viscous Saint-Venant system. In order to achieve the optimal decay rate, we also perform a detailed spectral analysis of our abstract problem under...

We consider the quad curl problem in smooth and non smooth domains of the space. We first give an augmented variational formulation equivalent to the one from [25] if the datum is divergence free. We describe the singularities of the variational space which correspond to the ones of the Maxwell system with perfectly conducting boundary conditions....

In this paper, we consider a damped wave equation with a dynamic boundary control. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case for the unit disk. Hence, we look for a polynomial decay...

We analyze the use of nonfitting meshes for simulating the propagation of electromagnetic waves inside the earth with applications to borehole logging. We avoid the use of parameter homogenization and employ standard edge finite element basis functions. For our geophysical applications, we consider a 3D Maxwell’s system with piecewise constant cond...

In this paper, the spectrum of the following fourth order problem \begin{equation*} \begin{cases} \Delta^2 u+\nu u=-\lambda \Delta u &\text{in } D_1,\newline u=\partial_r u= 0 &\text{on } \partial D_1, \end{cases} \end{equation*} where $D_1$ is the unit ball in ${\mathbb R}^N$, is determined for $\nu < 0$ as well as the nodal properties of the corr...

In \cite{WehbeRayleigh:06}, Wehbe considered the Rayleigh beam equation with two dynamical boundary controls and established the optimal polynomial energydecay rate of type $\dfrac{1}{t}$. The proof exploits in an explicit waythe presence of two boundary controls, hence the case of the Rayleigh beam damped by only one dynamical boundary control...

In the domain of field computation with the finite element method, choosing the mesh refinement is an important step to obtain an accurate solution. In order to evaluate the quality of the mesh, a posteriori error estimators are frequently used. In this communication we propose to analyze and to compare residual and equilibrated error estimators fo...

This paper deals with the stability analysis of a class of switched linear systems evolving on an arbitrary time domain. This class consists of a set of unstable linear continuous-time and unstable linear discrete-time subsystems. Using the time scale theory, some sufficient conditions are derived to guarantee the exponential stability of this clas...

An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak Gâteaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and an associa...

In this paper a penalized method and its approximation by �nite element method are proposed to solve Koiter's equations for a thin linearly elastic shell. In addition to existence and uniqueness results of solutions of the continuous and the discrete problems we derive some a priori error estimates. We are especially interested in the behavior of t...

We perform the analysis of the hp finite element approximation for the solution to singularly perturbed transmission problems, using Spectral Boundary Layer Meshes. In [12] it was shown that this method yields robust exponential convergence, as the degree p of the approximating polynomials is increased, when the error is measured in the energy norm...

We perform some hierarchical analyses of dissipative systems. For that purpose, we first propose a general abstract setting, prove a convergence result and discuss some stability properties. This abstract setting is then illustrated by significant examples of damped (acoustic) wave equations for which we characterize the family of reduced problems....

We consider 2 × 2 (first order) hyperbolic systems on networks subject to general transmission conditions and to some dissipative boundary conditions on some external vertices. We find sufficient but natural conditions on these transmission conditions that guarantee the exponential decay of the full system on graphs with dissipative conditions at a...

This paper is concerned with the theory of thermoelastic dipolar bodies which have a double porosity structure. In contrast with previous papers dedicated to classical elastic bodies, in our context the double porosity structure of the body is influenced by the displacement field, which is consistent with real models. In this setting, we show insta...

In this communication, an a posteriori equilibrated error estimator for the 3D Finite Element approximation of the eddy current problem is developed. Upper and lower bounds for the error are provided. Finally some numerical tests are performed in order to validate the theoretical results and check the efficiency of the estimator.

We consider the Laplace equation with a right-hand side concentrated on a curved fracture of class Cm+2 for some nonnegative integer m (i.e., a sort of Dirac mass). We show that the solution belongs to a weighted Sobolev space of order m, the weight being the distance to this fracture. Our proof relies on a priori estimates in a dihedron or a cone...

In this article, a guaranteed equilibrated error estimator is proposed for the harmonic magnetodynamic formulation of the Maxwell’s system. This system is recast into two classical potential formulations, which are solved by a finite element method. The equilibrated estimator is built starting from these dual problems, and is consequently available...