Pseudo-Parabolic KWC System

• The pseudo-parabolic KWC system is a phase-field model that augments traditional formulations with memory-type regularization to restore uniqueness and a gradient-flow structure.
• Recent studies confirm well-posedness by establishing existence, uniqueness, and regularity for both strong and weak solutions via coupled PDE analysis.
• The framework extends to optimal control scenarios, enabling efficient gradient algorithms and practical computational implementations for modeling grain boundary evolution.

The pseudo-parabolic Kobayashi–Warren–Carter (KWC) system constitutes a mathematically and physically rigorous phase-field model for planar grain boundary motion in materials science. Originating from the canonical KWC system by Kobayashi et al. (2000), the pseudo-parabolic extension resolves deficiencies of uniqueness and physical gradient-flow structure by embedding dissipation regularizations. The model involves two coupled PDEs for the orientation-order phase-field and the orientation angle, formulated as gradient flows of nonconvex BV-based energy functionals in Hilbert space with memory-type pseudo-parabolic regularization. Recent research has established comprehensive well-posedness—including existence, uniqueness, regularity, and continuous dependence—both for strong and weak solution frameworks, and has initiated optimal control theory for such systems (Antil et al., 16 Feb 2024, Antil et al., 11 Jun 2025, Mizuno, 26 Jul 2024).

1. Mathematical Formulation and System Definition

Let $\Omega \subset \mathbb{R}^N$ (with $1 \leq N \leq 4$) be a bounded domain with smooth boundary $\Gamma$, $T > 0$ a final time, and $Q = (0,T) \times \Omega$, $\Sigma = (0,T) \times \Gamma$. The unknowns are:

• $\eta(t,x)$: crystallinity (order) phase-field.
• $\theta(t,x)$: orientation angle field.

The pseudo-parabolic KWC system introduces regularization terms via parameters $\mu^2$ and $\nu^2$. Mobilities $\alpha_0(\cdot) > 0$ and $\alpha(\cdot) \geq 0$ are prescribed, as is a bulk perturbation $g(\cdot) = G'(\cdot)$ with $G \geq 0$. Volumetric forcings $u(t,x)$ and $v(t,x)$ are given. The system reads: $$\begin{aligned} &\partial_t \eta - \Delta (\eta + \mu^2 \partial_t \eta) + g(\eta) + \alpha'(\eta) \gamma_\varepsilon(\nabla\theta) = u, \quad \text{in } Q \ &\nabla(\eta + \mu^2 \partial_t \eta) \cdot n = 0, \quad \text{on } \Sigma \ &\eta(0,x) = \eta_0(x), \quad x \in \Omega \ &\alpha_0(\eta)\,\partial_t \theta - \nabla \cdot \left( \alpha(\eta) \frac{\nabla\theta}{\gamma_\varepsilon(\nabla\theta)} + \nu^2 \nabla \partial_t \theta \right) = v, \quad \text{in } Q \ &\left(\alpha(\eta) \frac{\nabla\theta}{\gamma_\varepsilon(\nabla\theta)} + \nu^2 \nabla \partial_t \theta\right) \cdot n = 0, \quad \text{on } \Sigma \ &\theta(0,x) = \theta_0(x), \quad x \in \Omega \end{aligned}$$ where $$\gamma_\varepsilon(y) = \sqrt{\varepsilon^2 + |y|^2}$$ is a regularization, typically $\varepsilon \geq 0$.

2. Energy Functional and Gradient-Flow Structure

The governing energy, the (nonconvex) Kobayashi–Warren–Carter energy, is given by

$$F_\varepsilon[\eta, \theta] = \frac{1}{2} \int_\Omega |\nabla\eta|^2\,dx + \int_\Omega G(\eta)\,dx + \int_\Omega \alpha(\eta) \gamma_\varepsilon(D\theta)$$

where $G' = g$ and the total variation measure $|D\theta|$ enters via the BV-functional. Formally, the pseudo-parabolic dynamics are described as: $$-\mathcal{A}_0(\eta) \frac{d}{dt} \begin{pmatrix} \eta \ \theta \end{pmatrix} = \nabla_{(\eta,\theta)} F_\varepsilon[\eta, \theta] + (u,v)$$ with self-adjoint dissipation operator

$$\mathcal{A}_0(\eta)\begin{bmatrix}\xi\\psi\end{bmatrix} = [ \xi - \mu^2\Delta\xi ;\, \alpha_0(\eta)\psi - \nu^2\Delta\psi ].$$

This gradient-flow-in-Hilbert-space structure distinguishes the pseudo-parabolic variant from the original parabolic KWC formulation.

3. Well-Posedness: Existence, Uniqueness, Regularity

Recent results (Antil et al., 16 Feb 2024, Mizuno, 26 Jul 2024, Antil et al., 11 Jun 2025) establish:

• Existence of strong solutions for smooth initial data: For $\eta_0, \theta_0 \in H^2(\Omega)$ (with homogeneous Neumann b.c.), $u \in L^\infty(Q)$, $v \in L^2(Q)$, there exists $\eta \in W^{1,2}(0,T;W_0) \cap L^\infty(Q)$ and $$\theta \in W^{1,2}(0,T; H^1(\Omega)) \cap L^\infty(0,T; W_0)$$ that satisfy the variational formulations.
• Existence and uniqueness of weak solutions: For initial data $$(\eta_0, \theta_0) \in [H^1(\Omega) \cap L^\infty(\Omega)] \times H^1(\Omega)$$, there exists a unique weak solution $(\eta, \theta)$ in

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

satisfying precise energy-dissipation inequalities (Mizuno, 26 Jul 2024).

• Regularization by pseudo-parabolic terms: The inclusion of $\mu^2\Delta\partial_t \eta$ and $\nu^2\Delta\partial_t \theta$ restores coercivity and enables uniqueness via $H^1$/$H^2$ control and Grönwall-based estimates.
• Continuous dependence: Solutions depend continuously on initial data, forcing terms, and regularization index $\varepsilon$,

$$J_n(t) \leq J_n(0)\exp\left(C\int_0^t\left[1+|\partial_t\eta|_H + |\partial_t\theta|_V\right] dr\right) + \Phi(\varepsilon_n - \varepsilon, u_n-u, v_n-v)$$

where $J_n$ is a solution difference energy defined explicitly.

4. Weak Solution Theory and Stability

For general $L^\infty$ initial data and forcing, one passes to a weak (variational) framework:

• Variational Formulation: For all $\varphi \in H^1(\Omega), \psi \in H^1(\Omega)$:
  • $\eta$ equation as an $L^2$ equality:

  $$(\partial_t\eta + g(\eta) + \alpha'(\eta) \gamma_\varepsilon(\nabla\theta), \varphi) + (\nabla(\eta + \mu^2 \partial_t\eta), \nabla\varphi) = (u,\varphi)$$ - $\theta$ equation as a variational inequality:

  $$(\alpha_0(\eta) \partial_t\theta, \theta-\psi ) + \nu^2 (\nabla\partial_t\theta, \nabla(\theta-\psi)) + \int_\Omega \alpha(\eta) \gamma_\varepsilon(\nabla\theta) \leq \int_\Omega \alpha(\eta) \gamma_\varepsilon(\nabla\psi) + (v,\theta-\psi)$$

• Energy-dissipation law:

$$C_0 \int_s^t \left( |\partial_t\eta|_{H^1}^2 + |\partial_t\theta|_{H^1}^2 \right) dr + F_\varepsilon(\eta(t), \theta(t)) \leq F_\varepsilon(\eta(s), \theta(s)) + \frac{1}{2} \int_s^t (|u|_{L^2}^2 + \frac{1}{\delta_*}|v|_{L^2}^2) dr$$

• Maximum principle and uniform boundedness: BV-type maximum principles ensure $L^\infty$ bounds for $\eta$, and for $\theta$ in the case of zero $v$.

5. Physical Interpretation and Variational Implications

The pseudo-parabolic regularization imparts both physical and analytical properties:

• Gradient-flow structure: The system now admits a classical gradient-flow interpretation with respect to the KWC energy in a Hilbert-space metric augmented by the pseudo-parabolic operator. This confers direct connection to physical dissipation and variational principles.
• Restoration of Well-Posedness: The original ($\mu = \nu = 0$) system lacks uniqueness and a direct variational gradient-flow connection; pseudo-parabolic regularizations overcome these deficits and allow standard existence and uniqueness theories to be applied.
• Interface drag mechanism: In physical terms, $\mu^2 \partial_t \eta$ and $\nu^2 \partial_t \theta$ act as viscous regularizations, conferring additional interface drag and memory effects—essential for accurate modeling of grain-boundary evolution.

6. Optimal Control and Algorithmic Aspects

The introduction of well-posed pseudo-parabolic KWC systems has enabled rigorous optimal control theory (Antil et al., 11 Jun 2025):

• Admissible controls: Restrict $(u,v)$ to $$U_{ad} = \{ (u,v) \in L^2(Q)^2 \mid \underline u \leq u \leq \overline u \}$$.
• Tracking cost functional:

$$J_\varepsilon(u,v) = \frac{M_\eta}{2} \int_0^T \|\eta - \eta_{ad}\|_{L^2}^2 + \frac{M_\theta}{2} \int_0^T \|\theta - \theta_{ad}\|_{L^2}^2 + \frac{M_u}{2} \int_0^T \|u\|_{L^2}^2 + \frac{M_v}{2} \int_0^T \|v\|_{L^2}^2$$

• Linearized state and adjoint equations: For incremental variables, linearized pseudo-parabolic PDEs are derived with coefficients frozen at the optimal state.
• First-order optimality conditions: Gradient characterization via adjoint variables yields projection operators for $u$ and explicit updates for $v$,

$$u^* = \mathrm{proj}_{[\underline u, \overline u]} \left(-\frac{L_u}{M_u} p^* \right), \quad v^* = -\frac{L_v}{M_v} z^*$$

• Gradient algorithms: Iterative schemes combining forward state, backward adjoint, and control projections are feasible.
• Semi-continuous dependence: The optimal control and value functionals inherit stability under data perturbation, facilitated by the unique solvability and continuous dependence of the underlying state system.

7. Comparative Analysis and Research Directions

• Contrast with classical models: The pseudo-parabolic KWC system overcomes the lack of uniqueness, physical gradient-flow correspondence, and regularity failures endemic to the original parabolic KWC formulation.
• Scope of applicability: Results extend to dimensions $N \leq 4$ with Lipschitz domains, admit $L^\infty$ data, and cover both strong and weak solution frameworks. A plausible implication is that the methods can be generalized to broader classes of BV-energy-gradient-flow PDEs in interface evolution.
• Current developments: Recent advances include the extension to weak initial data, the explicit formulation and analysis of optimal control problems, and detailed algorithmic prospects for computational realization.
• Open problems: Questions remain regarding further physical interpretation of pseudo-parabolic drag in experimentally observed grain boundary motion and extension to nonplanar or three-dimensional geometries.

In summary, the pseudo-parabolic KWC system establishes a comprehensive and physically consistent mathematical foundation for the modeling, analysis, and control of grain boundary dynamics, resolving longstanding issues in both theory and application.