Mesures semi-classiques et mesures de défaut
Method of replacing the variables for generalized symmetry of the D'Alembert equation.
Microlocal analysis and seismic imaging
We study certain Fourier integral operators arising in the inversion of data from reflection seismology.
Moving Dirichlet boundary conditions
This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second...
Neumann operator for wave equation in a half space and microlocal orders of singularities along the boundary
New singular solutions of Protter's problem for the 3D wave equation.
Nombres premiers et vibrations
Nonexistence of global solutions to some differential inequalities of the second order and applications.
Nonlinear oscillations of completely resonant wave equations
Non-Resonance for a Strongly Dissipative Wave Equation in Higher Dimensions.
Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system.
Numerical analysis of Eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays
The purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays. First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model.5 (2001) 537–572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue. The classical multi-fluid method consists then in...
Numerical comparisons of two long-wave limit models
The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations 16 (2003) 1039–1064; Pego and Quintero, Physica D 132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...
Numerical comparisons of two long-wave limit models
The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known (Paumond, Differential Integral Equations16 (2003) 1039–1064; Pego and Quintero, Physica D132 (1999) 476–496) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We study here numerically...
Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume
We consider the numerical solution, in two- and three-dimensional bounded domains, of the inverse problem for identifying the location of small-volume, conductivity imperfections in a medium with homogeneous background. A dynamic approach, based on the wave equation, permits us to treat the important case of “limited-view” data. Our numerical algorithm is based on the coupling of a finite element solution of the wave equation, an exact controllability method and finally a Fourier inversion for localizing...
Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume
We consider the numerical solution, in two- and three-dimensional bounded domains, of the inverse problem for identifying the location of small-volume, conductivity imperfections in a medium with homogeneous background. A dynamic approach, based on the wave equation, permits us to treat the important case of “limited-view” data. Our numerical algorithm is based on the coupling of a finite element solution of the wave equation, an exact controllability method and finally a Fourier inversion for...
Observability inequalities and measurable sets
This paper presents two observability inequalities for the heat equation over . In the first one, the observation is from a subset of positive measure in , while in the second, the observation is from a subset of positive surface measure on . It also proves the Lebeau-Robbiano spectral inequality when is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.
Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes
The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math.102 (2006) 413–462] to nonuniform meshes. Our results...
On a counterexample of Garofalo-Lin for a unique continuation of Schrödinger equation.
On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values
We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values with By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case of (P) in which the nonlinear term contains the sum . Under suitable conditions, we prove that the solution of converges to the solution of the corresponding...