[Paper Review] On the regularity of global attractors

Monica Conti, Vittorino Pata|ArXiv.org|Jan 23, 2009

Stability and Controllability of Differential Equations17 references3 citations

TL;DR

This paper introduces a novel technique to establish the optimal regularity of global attractors for dissipative dynamical systems without relying on uniform-in-time estimates. By leveraging exponential attraction in a higher regularity space, the authors prove that the global attractor for the strongly damped wave equation with critical nonlinearity is bounded in $ H^1 \times H^1 $, achieving optimal regularity through a bootstrap argument based on exponential convergence to a bounded set in $ \mathcal{H}^{1/4} $. The method avoids the need for uniform bounds on the solution’s higher-order norms, offering a more accessible route to regularity results.

ABSTRACT

This note is focused on a novel technique in order to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application of the abstract result, the semigroup generated by the strongly damped wave equation $$u_{tt}-Δu_t-Δu+ϕ(u)=f$$ with critical nonlinearity is considered, whose attractor is shown to possess the optimal regularity.

Motivation & Objective

• To develop a new technique for proving the regularity of global attractors in higher-order spaces without relying on uniform-in-time bounds.
• To address the longstanding challenge of establishing optimal regularity for attractors of dissipative systems, particularly when standard uniform estimates are unattainable.
• To demonstrate the applicability of the method to the strongly damped wave equation with critical growth nonlinearity.
• To show that exponential attraction in a higher regularity space suffices to infer boundedness of the attractor in that space, enabling a bootstrap argument for optimal regularity.

Proposed method

• Introduce a new criterion for attractor regularity based on exponential attraction in a higher-order Banach space $ \mathcal{V} $, where $ \mathcal{V} $ is compactly embedded in the energy space $ \mathcal{H} $.
• Establish an exponential decay estimate: $ \text{dist}_{\mathcal{H}}(S(t)\mathfrak{B}, \mathfrak{C}) \leq J(\|\mathfrak{B}\|_{\mathcal{H}}) e^{-\omega t} $, with $ \mathfrak{C} \subset \mathcal{V} $, for some $ \omega > 0 $ and increasing function $ J $.
• Apply a novel differential inequality framework (Theorem 3.2) that links exponential decay in $ \mathcal{H} $ to boundedness in $ \mathcal{V} $, using energy functionals and perturbation estimates.
• Use the transitivity of exponential attraction to propagate regularity from $ \mathcal{H}^{1/4} $ to $ \mathcal{H}^1 $, enabling a bootstrap argument.
• Leverage the structure of the strongly damped wave equation to derive exponential decay estimates for the associated solution operators via energy estimates and Sobolev embeddings.
• Apply Hölder and Sobolev inequalities to control nonlinear terms in the energy estimates, ensuring the required decay rates for the solution operators.

Experimental results

Research questions

• RQ1 Can the regularity of a global attractor be established without relying on uniform-in-time bounds in the higher-order norm?
• RQ2 Does exponential attraction in a higher regularity space imply boundedness of the global attractor in that space?
• RQ3 Can the method be applied to the strongly damped wave equation with critical nonlinearity to achieve optimal regularity?
• RQ4 Is it possible to bypass the need for uniform estimates by using only the asymptotic behavior of the semigroup?
• RQ5 How can the transitivity of exponential attraction be used to propagate regularity through a hierarchy of spaces?

Key findings

• The global attractor for the strongly damped wave equation with critical nonlinearity is bounded in $ \mathcal{H}^1 = H^1 \times H^1 $, achieving optimal regularity.
• The attractor is shown to be bounded in $ \mathcal{H}^{1/4} $ via exponential attraction and a novel differential inequality framework, without requiring uniform-in-time bounds.
• A bootstrap argument based on the $ \mathcal{H}^{1/4} $-regularity and subcritical structure of the problem establishes $ \mathcal{H}^1 $-boundedness in a single step.
• The method yields the exponential attraction estimate: $ \text{dist}_{\mathcal{H}}(S(t)\mathfrak{B}, B_{\mathcal{H}^1}(\varrho)) \leq J(\|\mathfrak{B}\|_{\mathcal{H}}) e^{-\omega t} $, with $ \varrho > 0 $, $ \omega > 0 $, and $ J \in \mathfrak{I} $.
• The result extends to $ H^2 \times H^2 $ under the assumption that $ \varphi \in C^1(\mathbb{R}) $, showing the method's robustness to higher regularity.