Caroline Fabre

ON THE DENSITY OF THE RANGE OF THE SEMIGROUP FOR SEMILINEAR HEAT EQUATIONS

We consider the semilinear heat equation u t − Δu + f(u) = 0 in a bounded domain Ω ⊂ R n , n ≥ 1, for t > 0 with Dirichlet boundary conditions u = 0 on ∂Ω × (0,∞). For T > 0 fixed we consider the map S(T) : C 0 (Ω) → C 0 (Ω) such that S(T )u 0 = u (x, T) where u is the solution of this heat equation with initial data u(x, 0) = u 0(x) and C 0(Ω) is the space of uniformly continuous functions on Ω that vanish on its boundary. When f is globally Lipschitz and for any T > 0 we prove that the range of S(T)is dense in C 0(Ω). Our method of proof combines backward uniqueness results, a variational approach to the problem of the density of the range of the semigroup for linear heat equations with potentials and a fixed point technique. These methods are similar to those developed by the authors in an earlier paper in the study of the approximate controllability of semilinear heat equations.

EXISTENCE FOR VISCOPLASTIC CONTACT WITH COULOMB FRICTION PROBLEMS

We present existence results in the study of nonlinear problem of frictional contact between an elastic-viscoplastic body and a rigid obstacle. We model the frictional contact both by a Tresca’s friction law and a regularized Coulomb’s law. We assume, in a first part, that the contact is bilateral and that no separation takes place. In a second part, we consider the Signorini unilateral contact conditions. Proofs are based on a time-discretization method, Banach and Schauder fixed point theorems. 2000 Mathematics Subject Classification: 74A55, 74B20, 35Q99, 49J40. 1. Introduction, notation, and main results. This paper deals with the analysis of nonlinear frictional contact problems between an elastic-viscoplastic body and a rigid obstacle. We present both cases of a bilateral contact between the two bodies and a unilateral contact (involving Signorini model) and we consider nonlinear friction law. Before stating the scientific context and our results, we first introduce some notation that will be used in the paper. Let Ω be a bounded and regular open set of R d with boundary Γ . We suppose that Γ is divided in three disjoint parts Γ = Γ1 ∪ Γ2 ∪ Γ3 ,w ithΓ1 being on nonzero measure. We denote by Sd (d = 2o r 3) the space of symmetric tensors of order d on R d and it is endowed with its natural scalar product. If ν is the unit exterior normal on the boundary Γ ,a nd ifv is a vector in R d ,w e write vν = v ·ν and vτ = v −vν ν the normal and tangential decomposition of the vector v. In as ame way, we writeσν = σν · ν and στ = σν − σν ν the normal and tangential components of the vector σν for a tensor σ . We consider the following spaces (repeated convention indexes is used):

A discussion on Neumann–Kelvin's model for progressive water waves

This article focuses on the role of the capillarity in the modelling and the analysis of progressive waves at the surface of the sea. It is proved that the classical Neumann–Kelvins model is fully unstable and should be modified in order to be mechanically reliable.

Uniqueness results, control and asymptotic analysis for slightly compressible fluids

We prove a unique continuation property for (weak) solutions of slightly compressible fluid equations. We deduce approximate controllability for such equations. We present the asymptotic analysis when thc penaltys coefficient tends to infinity in control problems.

Contrôle et analyse asymptotique d'équations de fluide peu compressible

We prove a unique continuation property for (weak) solutions of slightly compressible fluids equations. We deduce approximate controllability for such equations, and then we do the asymptotic analysis when the penaltys coefficient tends to infinity in controls problems.