A mathematical study of an elastic-viscous-plastic sea-ice model with the Kelvin-Voigt rheology

1. Problem Statement and Motivation

The article addresses the mathematical analysis of the Elastic-Viscous-Plastic (EVP) sea-ice model, a widely used numerical regularization of the classical viscous-plastic Hibler model. Although the EVP model is computationally efficient for climate simulations, its rigorous mathematical well-posedness (existence, uniqueness, and continuous dependence on initial data) remains challenging, particularly in the presence of nonlinear advection terms and without artificial cut-offs of the viscosity coefficients.

Challenges:

• Non-Newtonian Rheology: Sea ice is modeled as a non-Newtonian fluid with complex stress-strain relationships involving plasticity.
• Advection Term: The inclusion of the nonlinear advection term ($u \cdot \nabla u$) in the momentum balance complicates energy estimates and the proof of global existence.
• Viscosity Cut-off: Standard models often require a "cut-off" (regularization parameter $\epsilon$) of the strain rate to avoid singularities. The removal of this cut-off (limit $\epsilon \to 0$) is a major open problem for multidimensional models.
• Previous Limitations: In a previous work by the authors (arXiv:2505.03080), they proved well-posedness for a Voigt-regularized model where the regularization was applied to the constitutive relationship (evolution of the stress tensor). However, this approach required high regularity of the initial data and could not handle the advection term when removing the viscosity cut-off.

Objective:
The authors propose a new formulation where the Kelvin-Voigt regularization is applied directly to the momentum balance (velocity field) rather than to the evolution of the stress tensor. Their goal is to prove:

1. Local well-posedness for the model with the advection term.
2. Global well-posedness for the model without the advection term.
3. The possibility of removing the viscosity cut-off ($\epsilon \to 0$) in both cases.
4. The existence of solutions for less regular initial data compared to previous studies.

2. Mathematical Formulation

The study considers the EVP model on a 2D flat torus $\mathbb{T}^2$. The system consists of a momentum equation and an evolution equation for the stress tensor $\sigma$.

The Regularized System (Kelvin-Voigt in the Momentum Balance):
The authors introduce a regularization parameter $\alpha > 0$ and modify the momentum equation (1.6a/1.7a) to insert a term $-\alpha^2 \partial_t \Delta u$:
$$\partial_t(u - \alpha^2 \Delta u) + (u \cdot \nabla)u = \nabla \cdot \sigma + \text{Force terms}$$
$$\frac{1}{E} \partial_t \sigma + \text{nonlinear stress terms} = D(u)$$
where $D(u)$ is the strain rate tensor.

Regularization of the Strain Rate:
To handle the singularity in the stress-strain relationship, the magnitude of the strain rate $D = |D(u)|$ is regularized as $D_\epsilon = \sqrt{|D(u)|^2 + \epsilon^2}$. The proofs involve constructing solutions for $\epsilon > 0$ and subsequently taking the limit $\epsilon \to 0$.

3. Methodology

The proof strategy is based on a multi-stage approximation and a compactness argument:

1. Double Regularization:

   • Strain Rate: Use of $D_\epsilon$ to smooth the nonlinearity.
   • Momentum: Introduction of a higher-order regularization term $\beta^4 \partial_t \Delta^2 u$ (with $\beta > 0$) in the momentum balance to ensure sufficient regularity for the Galerkin approximation scheme.
   • Initial Data: Smoothing of the initial data $(u_0, \sigma_0)$ using a standard mollifier $\phi_\delta$.

2. Galerkin Approximation:

   • Construction of finite-dimensional approximate solutions $(u_N, \sigma_N)$ using Fourier modes.
   • Proof of local existence for these ODE systems via the Picard-Lindelöf theorem.

3. A-priori Estimates:

   • $L^\infty$ Estimate for Stress (Crucial Step): The authors derive a new $L^\infty$ bound for the stress tensor $\sigma$ (Lemma 2.3). By introducing a transformed variable $\tau = \sigma + \frac{P}{2}I_2$ and utilizing the damping structure of the stress evolution equation, they prove:
     $$\|\sigma(\cdot, t)\|_{L^\infty} \leq \|\sigma_0\|_{L^\infty} + 2P$$
     This bound is independent of the regularization parameters and is essential for closing higher-order estimates.
   • Energy Estimates: Derivation of $H^1$ and $H^2$ estimates for the velocity, as well as $H^1$ estimates for the stress. The $L^\infty$ bound for $\sigma$ enables the control of nonlinear terms involving the stress tensor.

4. Limit Passage:

   • Step 1: Limit $\beta, \delta \to 0$ to recover the solution for the system with $D_\epsilon$ but without the artificial $\beta$-regularization.
   • Step 2: Limit $\epsilon \to 0$ to remove the strain rate cut-off.
   • Compactness: Use of the Banach-Alaoglu theorem (for weak-* convergence) and the Aubin-Lions lemma (for strong convergence) to extract convergent subsequences.

5. Uniqueness and Continuous Dependence:

   • Proof of uniqueness by analyzing the difference between two solutions and applying the Grönwall inequality, utilizing the $L^\infty$ bound for the stress tensor.

4. Main Contributions and Results

Theorem 1.1 (Local Well-posedness with Advection):

• Result: Proof of the existence of a unique local-in-time strong solution for the advective Kelvin-Voigt-EVP model.
• Regularity: $u \in C([0, T]; H^2)$ and $\sigma \in C([0, T]; H^1) \cap C([0, T]; L^\infty)$.
• Significance: This is the first rigorous result for the multidimensional EVP model with advection where the viscosity cut-off has been removed ($\epsilon \to 0$). Previous estimates could not handle the advection term in the limit passage.

Theorem 1.2 (Global Well-posedness without Advection):

• Result: Proof of the existence of a unique global-in-time strong solution for the Kelvin-Voigt-EVP model when the advection term is omitted.
• Significance: Global existence holds for initial data with lower regularity than required in the authors' previous work (where regularization was on the stress tensor). Specifically, the velocity is regularized, allowing for $H^2$ initial data instead of requiring higher smoothness.

Handling Viscosity Cut-off:

• The article successfully justifies the limit $\epsilon \to 0$ (removal of the artificial cut-off of the viscosity coefficients) in the presence of advection (Theorem 1.1), a problem that remains open for the standard Hibler model.

Physical Interpretation:

• The Kelvin-Voigt regularization in the momentum balance is argued to be more physically intuitive than the regularization of the constitutive relationship, as it models the elastic-viscous behavior of the ice directly in the force balance.

5. Significance and Impact

• Mathematical Rigor: The article provides the first rigorous well-posedness theory for the EVP model with advection and without viscosity cut-offs in multiple dimensions.
• Overcoming Open Problems: It solves a major open problem regarding the limit $\epsilon \to 0$ in the presence of advection, which was previously unsolvable for Hibler and Voigt-EVP models.
• Improved Regularity Requirements: By shifting the regularization to the momentum equation, the authors lower the required regularity of the initial data, making the model applicable to a broader class of physical scenarios.
• New Analytical Tools: The derivation of the $L^\infty$ estimate for the stress tensor (Lemma 2.3) is a novel technical contribution that enables the closure of energy estimates in the presence of strong nonlinearities.
• Climate Modeling: These results provide a solid mathematical foundation for the numerical schemes used in large-scale climate simulations, ensuring that the models are mathematically grounded and stable under realistic conditions.

In summary, this work advances the mathematical theory of sea-ice dynamics by establishing robust well-posedness results for a physically motivated regularization of the EVP model and overcoming significant analytical hurdles related to nonlinearity, advection, and singular viscosity limits.