Well-Posedness Theory for a Nonconservative Burgers-Type System Arising in Dislocation Dynamics

Abstract

In this work we study a system of nonconservative Burgers type in one space dimension, arising in modeling the dynamics of dislocation densities in crystals. Starting from physically relevant initial data that are of a special form, namely nondecreasing, periodic plus linear functions, we prove the global existence and uniqueness of a solution in $H^1_{loc}(\mathbb R\times[0,+\infty))$ that preserves the nature of the initial data. The approach is made by adding some viscosity to the system, obtaining energy estimates, and passing to the limit for vanishing viscosity. A comparison principle is shown for this system as well as an application in the case of the classical Burgers equation.