Sharp bounds on the attractor dimensions for damped wave equations

We give 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⊂ 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 Schrödinger 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 non-linearity 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.