Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities

El Hajj, A.

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 21-35.

Dans ce papier, nous étudions le modèle de Groma et Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] qui décrit la dynamique des densités de dislocations. Il s'agit d'un modèle bidimensionnel où les densités de dislocations satisfont un système de deux équations de transport. Le champ de vitesse dans ce système est la contrainte de cisaillement du matériau, calculée à partir de l'équation de l'élasticité linéaire. Cette contrainte de cisaillement peut être liée aux densités de dislocations par certaines transformations de Riesz. En se basant sur des estimations de type commutateurs, nous montrons que ce modèle admet une unique solution locale pour toutes données initiales dans $C^r\left({ℝ}^2\right)\cap L^p\left({ℝ}^2\right)$ pour $r>1$ et $1<p<+\infty$, où $C^r\left({ℝ}^2\right)$ est l'espace Hölder–Zygmund.

In this paper, we study the model of Groma and Balogh [I. Groma, P. Balogh, Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation, Acta Mater. 47 (1999) 3647–3654] describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show that this model has a unique local-in-time solution corresponding to any initial datum in the space $C^r\left({ℝ}^2\right)\cap L^p\left({ℝ}^2\right)$ for $r>1$ and $1<p<+\infty$, where $C^r\left({ℝ}^2\right)$ is the Hölder–Zygmund space.

Classification : 54C70, 35L45, 35Q72, 74H20, 74H25
Mots clés : Cauchy's problem, Non-linear transport equations, Non-local transport equations, System of hyperbolic equations, Riesz transform, Hölder–Zygmund space, Dynamics of dislocation densities

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