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

Xin Liu, Marita Thomas, Edriss S. Titi

TL;DR

The paper analyzes a one-dimensional plastic Burgers equation $\partial_t u + \partial_x(u^2/2) - \partial_x\sigma =0$ with $\sigma \in \partial\psi(\partial_xu)$ and $\psi(e)=|e|$ as a toy model for Hibler’s sea-ice dynamics. By introducing a viscoplastic regularization $\psi_\varepsilon$, the authors construct global $H^2$-solutions and derive uniform energy-dissipation bounds, enabling compactness and a limit passage $\varepsilon\to0$ to obtain a BV-solution $(u,R)$. They prove that the limit stress $R$ lies in the subdifferential of the total variation functional $\Psi$ evaluated at $u$, i.e., $-\partial_xR\in\partial\Psi(u)$, using Minty-type arguments in dual spaces. Oleinik’s entropy condition is preserved in the limit, ensuring admissible Burger-type shocks and plateaus, while the BV framework captures the plastic rheology. This work provides rigorous BV-solutions for a 1D model that mirrors the momentum balance with plastic rheology in Hibler’s sea-ice model, offering a mathematically tractable route to analyze non-smooth stresses in advection-dominated systems.

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.

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

TL;DR

The paper analyzes a one-dimensional plastic Burgers equation with and as a toy model for Hibler’s sea-ice dynamics. By introducing a viscoplastic regularization , the authors construct global -solutions and derive uniform energy-dissipation bounds, enabling compactness and a limit passage to obtain a BV-solution . They prove that the limit stress lies in the subdifferential of the total variation functional evaluated at , i.e., , using Minty-type arguments in dual spaces. Oleinik’s entropy condition is preserved in the limit, ensuring admissible Burger-type shocks and plateaus, while the BV framework captures the plastic rheology. This work provides rigorous BV-solutions for a 1D model that mirrors the momentum balance with plastic rheology in Hibler’s sea-ice model, offering a mathematically tractable route to analyze non-smooth stresses in advection-dominated systems.

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 . For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit 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.
Paper Structure (14 sections, 16 theorems, 189 equations)

This paper contains 14 sections, 16 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Burgers equation, total variation flow, and very singular diffusion
  3. Outline and results of this work
  4. Existence of $\mathrm{BV}$-solutions
  5. Uniform a priori estimates
  6. First uniform a priori estimates from the energy-dissipation balance
  7. Further, improved estimates
  8. Limit passage $\varepsilon\to0$ in the weak formulation of \ref{['ibvp-approx']}
  9. Limit passage in Oleinik's entropy condition \ref{['unif-bd-ueps-Oleinik']}
  10. Characterization of the limit $R$
  11. Weak formulation in $V^*:=(L^\infty(0,T;L^\infty(\mathbb{R})\cap\mathrm{BV}(\mathbb{R})))^*$
  12. Characterization of the limit $R$ in $Y^*:=X\cap L^1(0,T;\mathrm{BV}(\mathbb{R}))$
  13. Existence of approximate solutions, Proof of Theorem \ref{['Thm-Ex-Approx-Sol']}
  14. Acknowledgements:

Key Result

Theorem 1.2

Assume that the initial datum $u_\mathrm{in} \in L^\infty(\mathbb R) \cap \mathrm{BV}(\mathbb R) \cap L^2(\mathbb R)$ satisfies the compatibility condition Then there exists a $\mathrm{BV}$-solution of Cauchy problem CauchyP-burgers-plastic of the plastic Burgers equation in the sense of Definition def:bv-sol.

Theorems & Definitions (29)

  • Definition 1.1: $\mathrm{BV}$-solution of the plastic Burgers equation
  • Theorem 1.2: Existence of $\mathrm{BV}$-solutions for the plastic Burgers equation
  • Theorem 2.1: Existence of approximate solutions
  • Proposition 2.2: Energy-dissipation balance and first uniform a priori estimates
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4: Improved estimates and Oleinik's entropy condition
  • proof
  • Theorem 2.5: Sim87CSSL
  • ...and 19 more