Parabolic KWC System Overview

• The parabolic KWC system is a set of nonlinear parabolic PDEs modeling grain boundary motion by coupling phase-field evolution with singular diffusion components.
• It employs pseudo-parabolic regularization to address well-posedness, ensuring solution uniqueness, improved time regularity, and continuous dependence on initial data.
• Numerical schemes for the system preserve energy dissipation and range integrity, facilitating accurate simulations of grain boundary dynamics in polycrystalline materials.

A parabolic KWC system refers to a class of nonlinear parabolic partial differential equations derived from the Kobayashi–Warren–Carter (KWC) phase-field model for describing grain boundary motion in polycrystalline materials. Such systems are mathematically characterized by the coupling of Allen–Cahn-type and singular or degenerate quasilinear diffusion equations, typically featuring variable-dependent mobilities and singular energy densities. Their study involves rigorous analysis of well-posedness, regularity, boundary conditions, energy dissipation, and optimal control, and spans both analytic and numerical domains.

1. Formulation of the Parabolic KWC System

A general parabolic KWC system (for $\varepsilon\ge 0$) in a bounded domain $\Omega \subset \mathbb{R}^N$ ($N=1,2,3,4$) with smooth boundary $\Gamma$ and time horizon $T>0$ reads: $$\begin{cases} \partial_t \eta - \Delta \eta + g(\eta) + \alpha'(\eta) \sqrt{\varepsilon^2 + |\nabla \theta|^2} = u, & \text{in } (0,T)\times\Omega, \ \alpha_0(\eta)\,\partial_t \theta - \operatorname{div} \left( \alpha(\eta) \frac{\nabla \theta}{\sqrt{\varepsilon^2 + |\nabla \theta|^2}} + \kappa \nabla \theta \right) = v, & \text{in } (0,T)\times\Omega, \ \nabla \eta\cdot n = 0,\quad (\alpha(\eta)\frac{\nabla\theta}{\sqrt{\varepsilon^2+|\nabla\theta|^2}}+\kappa\nabla\theta)\cdot n = 0, & \text{on } (0,T)\times\Gamma, \ \eta(0,x) = \eta_0(x),\quad \theta(0,x) = \theta_0(x), & x\in\Omega. \end{cases}$$ Here,

• $\eta$ is a phase-field (orientation-order) parameter,
• $\theta$ is the crystalline orientation angle,
• $g$ is a nonlinearity (with $G' = g$, $G\ge0$),
• $\alpha$ and $\alpha_0$ are Lipschitz, bounded below by positive constants, representing mobility functions,
• $\kappa>0$ is a regularizing parameter,
• $u$ and $v$ are external forcings.

This system is the $L^2$-gradient flow of the regularized KWC energy functional,

$$\mathcal F_\varepsilon[\eta,\theta] = \frac12\int_\Omega |\nabla \eta|^2\,dx + \int_\Omega G(\eta)\,dx + \int_\Omega \alpha(\eta) \sqrt{\varepsilon^2 + |\nabla \theta|^2}\,dx.$$

In the singular limit $\varepsilon\to0$, the energy term $\int_\Omega \alpha(\eta)|\nabla\theta|\,dx$ introduces degenerate diffusion and possible discontinuities in $\theta$.

2. Well-posedness and Regularity Theory

Nontrivial issues in the analysis of parabolic KWC systems arise from the coupling of nonlinear (often singular) diffusion with variable-dependent mobility and the lack of uniform convexity in the energy density for $\varepsilon=0$.

Well-posedness for variable-dependent mobilities: The breakdown of standard uniqueness proofs for non-constant $\alpha_0(\eta)$ is addressed by pseudo-parabolic regularization. Key results show that introducing higher-order time derivatives (via pseudo-parabolic terms $-\mu^2\Delta\partial_t\eta$ and $-\nu^2\Delta\partial_t\theta$) ensures the necessary $H^1$-regularity in time for $\theta$, enabling uniqueness and continuous dependence (Mizuno et al., 7 Dec 2025, Mizuno, 26 Jul 2024). This regularization is removed in the limit to recover solutions to the original parabolic KWC system.

Existence and uniqueness: Under suitable assumptions on data and coefficients, there exists a unique solution pair $(\eta, \theta)$ with

$$\eta \in W^{2,2}(0,T;L^2(\Omega)) \cap W^{1,\infty}(0,T;H^1(\Omega)),\quad \theta \in W^{1,2}(0, T; H^1(\Omega)) \cap L^\infty(0,T; H^1(\Omega)),$$

and the mapping $$(\varepsilon, \eta_0, \theta_0, u, v)\mapsto (\eta,\theta)$$ is continuously dependent (Mizuno et al., 7 Dec 2025).

Weak solution framework: For weaker initial data, existence and uniqueness of weak solutions holds in

$$\eta \in W^{1,2}(0,T;H^1(\Omega)) \cap L^\infty((0,T)\times \Omega), \quad \theta \in W^{1,2}(0,T;H^1(\Omega))$$

with the energy-dissipation inequality holding for all $0\leq s\leq t\leq T$ (Mizuno, 26 Jul 2024).

Boundary conditions and dynamic boundaries: The literature also treats dynamic boundary conditions modeling heat and phase exchanges (e.g., transmission conditions $\eta_\Gamma$, $\theta_\Gamma$), with the main interest in reconciling continuity of traces with possible discontinuities induced by the singular diffusion (see (Nakayashiki et al., 2023, Kubota et al., 2020)).

3. Energy Structure and Dissipation

Central to the analysis is the variational and dissipative structure:

• Energy functional decrease: For all sufficiently regular solutions, the total free energy (including interfacial and singular terms) is nonincreasing:

$$t \mapsto \mathcal F_\varepsilon[\eta(t),\theta(t)]$$

is nonincreasing and right-continuous (Okumura et al., 20 Jun 2025, Mizuno, 26 Jul 2024, Nakayashiki et al., 2023).

• Energy-dissipation inequality:

$$\int_0^t \big( \|\partial_t \eta(s)\|_{L^2}^2 + \|\alpha_0(\eta(s))\partial_t \theta(s)\|_{L^2}^2 \big)ds + \mathcal F_\varepsilon[\eta(t),\theta(t)] \le \mathcal F_\varepsilon[\eta(0),\theta(0)],$$

quantifies the dissipative evolution (Nakayashiki et al., 2023).

• Structural properties in numerical schemes: Discrete maximum principles and discrete energy dissipation are preserved in carefully designed schemes for 1D systems (Okumura et al., 20 Jun 2025).

4. Boundary Conditions and Transmission Relations

Parabolic KWC systems are often posed with homogeneous Neumann boundary conditions. For applications such as grain boundary motion in polycrystals, dynamic boundary conditions are essential to model interfacial exchanges:

• Transmission conditions: On the boundary $\Gamma$,

$$\eta|_\Gamma = \eta_\Gamma,\quad \theta|_\Gamma = \theta_\Gamma,$$

supplemented by dynamic evolution for $\eta_\Gamma$ and $\theta_\Gamma$ encoding heat or order-parameter exchange (Nakayashiki et al., 2023, Kubota et al., 2020).

• Singular diffusion and boundary interaction: The degenerate (total variation) gradient flow structure introduces a mathematical conflict between required continuity at $\Gamma$ and the potential for interior discontinuity in $\theta$. The main resolution—expressed via subdifferential calculus—is to encode both effects in a maximal monotone operator $\partial \Phi_0(\alpha(\eta); \cdot)$ acting on the trace pair $[\theta, \theta_\Gamma]$ (Nakayashiki et al., 2023).

5. Numerical Discretization and Structure Preservation

Development of numerical schemes for parabolic KWC systems places critical focus on preserving structural features:

• Range preservation: Discrete solutions satisfy $0\leq \eta \leq 1$ and $|\theta| \leq \text{const}$ for all times and grid points, by maximum-principle arguments (Okumura et al., 20 Jun 2025).
• Energy dissipation: The fully implicit structure-preserving scheme guarantees that a discrete analog of the energy functional decreases monotonically with each time step.
• Convergence and error estimates: Under regularity assumptions, the schemes are first-order accurate in both temporal and spatial discretizations (up to Hölder regularity corrections), with error estimates of the form

$$\|\mathbf{H}^{(j)} - \eta(j\Delta t)\|_{\ell^2} + \|\mathbf{\Theta}^{(j)} - \theta(j\Delta t)\|_{\ell^2} \le C\big((\Delta x)^\sigma + (\Delta t)^\sigma + \Delta x + \Delta t + (\Delta x)^2\big)$$

(Okumura et al., 20 Jun 2025).

6. Periodic, Optimal Control, and Pseudo-Parabolic Variants

Time-periodic solutions: Existence of nontrivial time-periodic solutions is established without the compromised assumption of constant mobility, by variational methods and Mosco/T-convergence to pass to the singular case (Kubota et al., 2023).

Optimal control: Both state-system and optimality system theory are developed for 1D and multidimensional KWC systems, with physically meaningful controls (internal and boundary) and cost functionals reflecting discrepancies from target order and orientation fields. First-order necessary conditions involve adjoint pseudo-parabolic equations and multivalued inclusions in the singular limit (Antil et al., 11 Jun 2025, Kubota et al., 2020).

Pseudo-parabolic regularization: Uniqueness and higher time regularity are ensured via pseudo-parabolic modifications—incorporating higher-order time-derivatives into the mobility operator—facilitating compactness and convergence properties essential for both analysis and optimization (Mizuno, 26 Jul 2024, Antil et al., 11 Jun 2025).

7. Mathematical and Modeling Implications

• The parabolic KWC system establishes a well-posed, physically realistic model for grain-boundary dynamics with variable-dependent mobilities and possible singular energies.
• The existence of nonunique solutions (in certain contexts, for singular limits or measure-valued data) reflects intrinsic selection ambiguities typical of degenerate dissipative systems, emphasizing the need for additional admissibility or entropy dissipation criteria (Antil et al., 11 Jun 2025).
• The robust regularity theory under pseudo-parabolic approximation is crucial for the mathematical foundation of sharp-interface limits and computational stability in components involving singular degenerate diffusion (Mizuno et al., 7 Dec 2025, Mizuno, 26 Jul 2024).
• Recent advances have resolved major open problems concerning existence, uniqueness, and structure-preserving discretization for parabolic KWC systems, paving the way for rigorous optimal control and large-scale simulation frameworks in materials science.

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Key references:

• Existence, uniqueness, and regularity with variable-dependent mobilities: (Mizuno et al., 7 Dec 2025, Mizuno, 26 Jul 2024).
• Structure-preserving schemes and energy dissipation: (Okumura et al., 20 Jun 2025).
• Dynamic boundary conditions and singular diffusion: (Nakayashiki et al., 2023, Kubota et al., 2020).
• Periodic solutions free of compromised assumptions: (Kubota et al., 2023).
• Optimal control of pseudo-parabolic KWC systems: (Antil et al., 11 Jun 2025).