Sharp bounds on the attractor dimensions for damped wave equations

TL;DR

This paper derives explicit bounds on the fractal dimension of attractors for damped wave equations across various dimensions, revealing the influence of damping and nonlinearity on attractor complexity.

Contribution

It provides the first explicit estimates of attractor dimensions for damped hyperbolic equations in arbitrary dimensions, including sharp bounds and the role of logarithmic corrections.

Findings

Bounds of order γ^{-d} for attractor dimensions in all dimensions

Logarithmic correction in 1D case appears inevitable

Lyapunov dimension scales as γ^{-1} in all dimensions

Abstract

We give the explicit estimates of order $\gamma^{-d}$ (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 $\Omega\subset \mathbb R^d$, $d\ge1$ with linear damping coefficient $\gamma>0$. The key ingredient in the proof for $d\ge3$ is Lieb's bound for the $L_p$-norms of systems with orthonormal gradients based on the Cwikel--Lieb--Rozenblum (CLR) inequality for negative eigenvalues of the Schr\"odinder 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 $d\ge2$ 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 $\gamma^{-1}$ in all spatial dimensions $d\ge1$.