Ahmad El Hajj

Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system.

GLOBAL CONTINUOUS SOLUTIONS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonal hyperbolic system appears naturally in the modeling of the dynamics of dislocation densities.

Direct algorithms for solving some inverse source problems in 2D elliptic equations

This paper deals with the resolution of some inverse source problems in the 2D elliptic equation from Cauchy data. Two types of sources are considered, pointwise sources and sources having compact support within a finite number of small subdomains. An identification direct algorithm, based on an algebraic approach, is proposed. This is a new result, as far as we know, except in the case which is already considered in El Badia and Ha-Duong (2000 Inverse Problems 16 651?63).

Stability estimates for an inverse source problem of Helmholtz’s equation from single Cauchy data at a fixed frequency

This paper deals with an inverse pointwise source problem for the Helmholtz equation in the three-dimensional case from single Cauchy data at a fixed frequency. Stability estimates of locations, intensities and moments for monopolar and dipolar sources are established.

UNIQUENESS RESULTS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

We study the uniqueness of solutions to diagonal hyperbolic systems in one spatial dimension and we present two uniqueness results. First, we establish a global existence and uniqueness theorem for continuous solutions to strictly hyperbolic systems. Second, we establish a global existence and uniqueness theorem for Lipschitz continuous solutions to hyperbolic systems that need not be strictly hyperbolic. Furthermore, an application is presented for one-dimensional flows in isentropic gas dynamics.

Homogenization of dislocation dynamics

In this paper we consider the dynamics of dislocations with the same Burgers vector, contained in the same glide plane, and moving in a material with periodic obstacles. We study two cases: i) the particular case of parallel straight dislocations and ii) the general case of curved dislocations. In each case, we perform rigorously the homogenization of the dynamics and predict the corresponding effective macroscopic elasto-visco-plastic flow rule.