Short time existence and uniqueness in H\"older spaces for the 2D dynamics of dislocation densities

TL;DR

This paper proves local-in-time existence and uniqueness of solutions for a 2D dislocation density model using H"older spaces, advancing understanding of dislocation dynamics in materials.

Contribution

It establishes the first local existence and uniqueness results for the 2D dislocation density model in H"older spaces, utilizing commutator estimates.

Findings

Unique local-in-time solutions exist for initial data in $C^r$ and $L^p$ spaces.

The model's velocity field relates to Riesz transforms of dislocation densities.

Mathematical framework extends previous results to H"older space setting.

Abstract

In this paper, we study the model of Groma and Balogh 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 thatthis model has a unique local-in-time solution corresponding to any initial datum in the space $C^r(\R^2)\cap L^p(\R^2)$ for $r>1$ and $1<p<+\infty$, where $C^r(\R^2)$ is the H\"older-Zygmund space.