Existence and uniqueness for a nonlinear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities
摘要
We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a one-dimensional system of a parabolic equation and a first order Hamilton-Jacobi equation that are coupled together. We show the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial data. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of an entropy solution of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of ``bounded from below'' solutions. In order to prove these results, we use a relation between scalar conservation laws and Hamilton-Jacobi equations, mainly to get some gradient estimates. This study takes place on , and on a bounded domain with suitable boundary conditions.
引用
@article{arxiv.math/0703783,
title = {Existence and uniqueness for a nonlinear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities},
author = {Hassan Ibrahim},
journal= {arXiv preprint arXiv:math/0703783},
year = {2007}
}