2024/09/27 by Alexei Ilyin, Ilyin, A. A., Anna Kostianko +3 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2409.18801
openalex publication_date 2024/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give the explicit estimates of order γ-d (with logarithmic correction in the 1D case) for the fractal dimension of the attractor of the damped hyperbolic equation (or system) in a bounded domain Ω⊂ \mathbb Rd, d≥1 with linear damping coefficient γ>0. The key ingredient in the proof for d≥3 is Lieb's bound for the Lp-norms of systems with orthonormal gradients based on the Cwikel--Lieb--Rozenblum (CLR) inequality for negative eigenvalues of the Schrödinder operator. The case d=1 is simpler, but contains a logarithmic correction term that seems to be inevitable. The 2D case is more difficult and is strongly based on the Strichartz-type estimates for the linear equation. Lower bounds of the same order for the dimension of the attractor are also obtained for a damped hyperbolic system with nonlinearity containing a small non-gradient perturbation term, meaning that in this case our estimates are optimal for d≥2 and contain a logarithmic discrepancy for d=1. Estimates for the various dimensions (Hausdorff, fractal, Lyapunov) of the attractor in purely gradient case are also given. We show, in particular, that the Lyapunov dimension of a non-trivial attractor is of the order γ-1 in all spatial dimensions d≥1.