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Global well-posedness of the elastic-viscous-plastic sea-ice model with the inviscid Voigt-regularisation

Daniel W. Boutros, Xin Liu, Marita Thomas, Edriss S. Titi

TL;DR

This work provides a rigorous global well-posedness theory for a Voigt-regularised elastic-viscous-plastic (EVP) sea-ice model on the periodic two-torus. The authors introduce an inviscid Voigt regularisation of the stress evolution, coupled with a regularised strain-rate $\mathcal{D}_\varepsilon$, and prove the existence and uniqueness of global strong solutions, obtaining detailed a priori bounds that grow super-exponentially in time. The analysis proceeds via a Galerkin approximation to an intermediate system with $\mathcal{D}_\varepsilon$, followed by a careful passage to the limits $N\to\infty$ and $\varepsilon\to0$, showing stability and convergence of nonlinear terms, including ocean and wind stress couplings. The results justify the use of Voigt regularisation for numerical implementation and provide a solid mathematical foundation for EVP-type sea-ice models, with potential extensions to weak solutions and other regularisations of the strain rate.

Abstract

In this paper, we initiate the rigorous mathematical analysis of the elastic-viscous-plastic (EVP) sea-ice model, which was introduced in [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849-1867]. The EVP model is one of the standard and most commonly used dynamical sea-ice models. We study a regularized version of this model. In particular, we prove the global well-posedness of the EVP model with the inviscid Voigt-regularisation of the evolution equation for the stress tensor. Due to the elastic relaxation and the Voigt regularisation, we are able to handle the case of viscosity coefficients without cutoff, which has been a major issue and a setback in the computational study and analysis of the related Hibler sea-ice model, which was originally introduced in [W. D. Hibler, J. Phys. Oceanogr., 9, 4 (1979), 815-846]. The EVP model shares some structural characteristics with the Oldroyd-B model and related models for viscoelastic non-Newtonian complex fluids.

Global well-posedness of the elastic-viscous-plastic sea-ice model with the inviscid Voigt-regularisation

TL;DR

This work provides a rigorous global well-posedness theory for a Voigt-regularised elastic-viscous-plastic (EVP) sea-ice model on the periodic two-torus. The authors introduce an inviscid Voigt regularisation of the stress evolution, coupled with a regularised strain-rate , and prove the existence and uniqueness of global strong solutions, obtaining detailed a priori bounds that grow super-exponentially in time. The analysis proceeds via a Galerkin approximation to an intermediate system with , followed by a careful passage to the limits and , showing stability and convergence of nonlinear terms, including ocean and wind stress couplings. The results justify the use of Voigt regularisation for numerical implementation and provide a solid mathematical foundation for EVP-type sea-ice models, with potential extensions to weak solutions and other regularisations of the strain rate.

Abstract

In this paper, we initiate the rigorous mathematical analysis of the elastic-viscous-plastic (EVP) sea-ice model, which was introduced in [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849-1867]. The EVP model is one of the standard and most commonly used dynamical sea-ice models. We study a regularized version of this model. In particular, we prove the global well-posedness of the EVP model with the inviscid Voigt-regularisation of the evolution equation for the stress tensor. Due to the elastic relaxation and the Voigt regularisation, we are able to handle the case of viscosity coefficients without cutoff, which has been a major issue and a setback in the computational study and analysis of the related Hibler sea-ice model, which was originally introduced in [W. D. Hibler, J. Phys. Oceanogr., 9, 4 (1979), 815-846]. The EVP model shares some structural characteristics with the Oldroyd-B model and related models for viscoelastic non-Newtonian complex fluids.
Paper Structure (15 sections, 6 theorems, 83 equations, 1 table)

This paper contains 15 sections, 6 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Formulation of the EVP model
  3. The linear ill-posedness of the unregularised EVP model
  4. Main result of this paper
  5. Overview of the literature
  6. Preliminaries
  7. Global solutions to the intermediate system \ref{['sys:simplifiedmodel']}
  8. The Galerkin approximation scheme
  9. Uniform-in-$(N, \epsilon)$ estimates
  10. Well-posedness of the intermediate system \ref{['sys:simplifiedmodel']}
  11. Proof of Theorem \ref{['regularisedevplimit']}
  12. Discussion and conclusion
  13. About the advection term
  14. Using the original strain rate
  15. Different regularisations of the strain rate

Key Result

Theorem 1.2

Let $u_0 \in H^2 (\mathbb{T}^2)$ and $\sigma_0 \in H^3 (\mathbb{T}^2)$ (which is symmetric) be given. Moreover, we assume that $P > 0$ is a given constant, $U_a, U_w, H_0 \in C([0,T]; H^4 (\mathbb{T}^2))$ and we let $\theta \in \left[0, \frac{\pi}{4} \right]$. The Voigt-EVP system sys:limitEVP has a In addition, we have the estimate where $C > 0$ is a constant, depending on $\mathcal{E}, \alpha,

Theorems & Definitions (12)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Remark 1.5
  • Lemma 2.1: Brézis-Gallouët–Wainger inequality
  • Lemma 2.2: Logarithmic Grönwall inequality
  • Lemma 2.3
  • proof
  • Lemma 2.4: Invariance of the symmetry
  • ...and 2 more