Regularity and stability of two coupled Euler-Bernoulli equations with a localized singular structural damping

1. Problem Statement

The paper investigates the long-term behavior (regularity and stability) of a system of two coupled Euler-Bernoulli plate equations subject to localized singular structural damping.

The system is defined on a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) with smooth boundary $\Gamma$:
$$\begin{cases} \partial_{tt} u + d \Delta^2 u - \text{div}(a(x) \nabla(\partial_t u + \partial_t v)) = 0 \\ \partial_{tt} v + c \Delta^2 v - \text{div}(a(x) \nabla(\partial_t u + \partial_t v)) = 0 \end{cases}$$
subject to clamped boundary conditions ($u = \partial_\nu u = v = \partial_\nu v = 0$) and initial data.

Key Features of the Model:

• Coupling: The damping term couples the velocities of the two plates ($\partial_t u + \partial_t v$). This is a "simultaneous" damping mechanism.
• Localization: The damping coefficient $a(x)$ is non-negative, supported on a proper open subset $\omega \subset \Omega$ (it vanishes outside $\omega$), rather than being global.
• Singular Nature: The damping matrix associated with the velocity vector is singular (not invertible), which complicates the analysis compared to standard global damping.
• Propagation Speeds: The plates have distinct flexural rigidity constants ($c \neq d$). The authors note that if $c=d$, the system becomes unstable due to a decoupled mode that does not dissipate energy.

The primary questions addressed are:

1. Does the associated $C_0$-semigroup decay to zero, and at what rate (exponential vs. polynomial)?
2. What is the regularity class of the semigroup (analytic, Gevrey, or merely differentiable)?

2. Methodology

The authors employ semigroup theory combined with advanced energy methods and multiplier techniques. The analysis is divided into two main parts: establishing Gevrey regularity under smooth damping assumptions and proving exponential stability under weaker (bounded) damping assumptions.

A. Well-posedness and Strong Stability

• The system is formulated as an abstract Cauchy problem $\dot{Z} = \mathcal{A}Z$ in the energy space $\mathcal{H} = (H^2_0(\Omega))^2 \times (L^2(\Omega))^2$.
• Using the Lumer-Phillips theorem, they prove the operator $\mathcal{A}$ generates a $C_0$-semigroup of contractions.
• Strong Stability: Proven by showing the resolvent set contains the imaginary axis ($\sigma(\mathcal{A}) \cap i\mathbb{R} = \emptyset$) using a unique continuation argument and the fact that $c \neq d$.

B. Proof of Gevrey Regularity (Theorem 1.1)

• Assumptions: The damping coefficient $a(x)$ is $C^2(\bar{\Omega})$ and satisfies specific structural constraints near the boundary of the support $\omega$ (specifically $|\nabla a|^4 \leq M a^3$). The support $\omega$ satisfies a geometric control condition (GC).
• Technique: The proof relies on Taylor's sufficient condition for Gevrey regularity, which requires establishing a resolvent estimate of the form $\|(i\lambda I - \mathcal{A})^{-1}\| = O(|\lambda|^{-\gamma})$ for large $\lambda$.
• Decomposition Strategy: To handle the localized damping, the authors decompose the velocity variables into a "smooth" part ($v_1, z_1$) and a "less smooth" part ($v_2, z_2$).
  • The smooth parts are estimated using standard elliptic regularity.
  • The less smooth parts are handled using interpolation inequalities and the specific structural constraints on $a(x)$.
• Propagation of Regularity: Since damping is localized, the authors use first-order multipliers (vector fields $m_j(x) = x - x_j^0$) and cut-off functions to propagate the dissipation from the region $\omega$ to the entire domain $\Omega$.
• Handling Singularity: A critical challenge is the singular damping matrix. The authors carefully combine estimates for the two plates, leveraging the distinct constants $c$ and $d$ to cancel out undesirable terms and propagate the damping effect.

C. Proof of Exponential Stability (Theorem 1.2)

• Assumptions: The damping coefficient $a(x)$ is only required to be in $L^\infty(\Omega)$ and bounded below by a positive constant on $\omega$. No smoothness is assumed.
• Technique: The proof uses the Arendt-Batty/Lyubich-Vu criterion (or similar resolvent criteria) which states that a bounded semigroup is exponentially stable if the resolvent is bounded on the imaginary axis.
• Adaptation: The authors adapt the multiplier method from the regularity proof but replace the smooth damping coefficient $a(x)$ with a smooth cut-off function $\eta$ that approximates the support of the damping. This allows them to bypass the need for derivatives of $a(x)$ while still capturing the localized energy dissipation.

3. Key Contributions and Results

Main Theorems

1. Theorem 1.1 (Gevrey Regularity):

   • Under the assumption that $a(x)$ is $C^2$ and satisfies structural constraints, the semigroup is of Gevrey class $s$ for every $s > 5/2$.
   • This implies the semigroup is infinitely differentiable for $t > 0$ but not analytic.
   • The resolvent satisfies $\|(i\lambda I - \mathcal{A})^{-1}\| \leq C |\lambda|^{-2/5}$.
   • Significance: This result matches the regularity established for a single Euler-Bernoulli plate with localized structural damping (previously shown in [15]), despite the added complexity of the coupled, singular system.

2. Theorem 1.2 (Exponential Stability):

   • If $a(x) \in L^\infty(\Omega)$ and is strictly positive on $\omega$ (satisfying the geometric condition), the energy decays exponentially: $E(t) \leq C_0 e^{-\mu t} E(0)$.
   • Significance: This extends stability results to non-smooth damping coefficients, which is physically more realistic but mathematically more challenging.

Comparative Insights

• Contrast with Wave Equations: The authors highlight a crucial difference between plate equations and wave equations. For coupled wave equations with localized simultaneous damping, the decay is typically polynomial (due to lack of differentiability of the semigroup). In contrast, the Euler-Bernoulli system here exhibits exponential decay even with localized damping, attributed to the "structural" (square-root type) nature of the damping which preserves higher regularity.
• Necessity of Distinct Speeds: The paper reiterates that $c \neq d$ is essential. If $c=d$, a mode exists that is not damped, leading to instability.

4. Significance

• Mathematical Rigor: The paper provides a rigorous analysis of a singular, coupled hyperbolic system with localized damping, a class of problems that are notoriously difficult due to the lack of global dissipation and the singularity of the coupling matrix.
• Optimality: The Gevrey class $s > 5/2$ is shown to be consistent with the single-plate case, suggesting that the coupling does not degrade the regularity properties of the system further than the localization of the damping itself.
• Physical Relevance: The results apply to real-world engineering structures (e.g., composite plates or sandwich structures) where damping is often localized (e.g., via piezoelectric patches or viscoelastic layers) and material properties may vary.
• Methodological Advancement: The decomposition of variables and the specific handling of the singular damping matrix via multiplier techniques offer a robust framework for analyzing similar coupled systems with localized controls.

In summary, the paper successfully demonstrates that two coupled Euler-Bernoulli plates with localized singular structural damping possess strong regularity (Gevrey class) and exponential stability, provided the propagation speeds are distinct and the damping support satisfies a geometric control condition.