Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics

TL;DR

This paper investigates a one-dimensional plastic Burgers equation with a multivalued stress term, modeling sea-ice dynamics, and establishes existence of solutions through regularization and singular limit analysis.

Contribution

It introduces a regularized approach to the plastic Burgers equation and proves the existence of BV solutions in the singular limit, linking stress limits to total variation subdifferentials.

Findings

Existence of global BV solutions for the regularized viscoplastic Burgers equation.

Convergence of regularized solutions to a BV solution of the plastic Burgers equation.

Identification of the stress limit as an element of the subdifferential of total variation.

Abstract

We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation functional. Our study highlights that the interplay of the advection term with the stresses given by the multivalued 1-Laplacian is a crucial feature of this model. Eventhough it is an interesting model in itsef, it can also be regarded as a one-dimensional version of the momentum balance of Hibler's model for sea-ice dynamics. Therein, the stress tensor is given by a term with similar properties as the 1-Laplacian in order to account for plastic effects of the ice. For our analysis we start out from a viscoplastic Burgers equation, i.e., a suitably regularized version of the plastic Burgers equation with a small regularization parameter $\varepsilon>0$. For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit $\varepsilon\to0$ we deduce the existence of a BV-solution for the plastic Burgers equation. In addition we show that the term arising as the limit of the regularized stresses is indeed related to an element of the subdifferential of the total variation functional.