Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system

Abstract

We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equations is written on a bounded interval with Dirichlet conditions and requires a special attention to the boundary. The proof of existence and uniqueness is done under the use of two main tools: a certain comparison principle on the gradient of the solution, and a parabolic Kozono-Taniuchi inequality.

Global existence of solutions to a singular parabolic/Hamilton–Jacobi coupled system with Dirichlet conditions

“…Concerning system (8), (9) and (10) we have the following global existence and uniqueness result: Theorem 1.3 (Global existence of smooth solutions for the regularized system, [6]). Let ρ ε,0 , κ ε,0 ∈ C ∞ (Ī ) satisfying the compatibility conditions:…”

“…Soit (ρ 0 , κ 0 ) une donnée initiale sur I satisfaisant (4), (5). Alors il existe une fonction (ρ, κ) telle que pour tout T > 0, (ρ, κ) ∈ (C(I T )) 2 avec ρ ∈ C(I T ), solution de (1), (6), avec les conditions initiales (2), et les conditions de Dirichlet au bord (3). L'idée de la preuve du Théorème 1 consiste à considérer une régularization parabolique (8) du système (1), en ajoutant une petite viscosité ε > 0.…”

Dynamics of Dislocation Densities in a Bounded Channel. Part II: Existence of Weak Solutions to a Singular Hamilton–Jacobi/Parabolic Strongly Coupled System

“…As we have already mentioned, we will use a parabolic regularization of (1.3), and a result of global existence of this regularized system from [17] (see Theorem 3.1). In order to use this result, we need to give a special attention to the conditions on the initial data of the approximated system 0 and 0 (see (3.1)-(3.3)).…”

“…The method of the proof consists in considering first a parabolic regularization of the full system, and then passing to the limit. For this regularized system, a result of global existence and uniqueness of a solution has been given in [17]. We show some uniform bounds on this solution which uses in particular an entropy estimate for the densities.…”

Derivation and study of dynamical models of dislocation densities

“…Even the regularized system (23) is still difficult to study, because of the division by κ y (see [10]). But for the regularized system (23), it is possible to replace M by…”