Sharp bounds on the attractor dimensions for damped wave equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929518301675520 |
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| author | Ilyin, A. A. Kostianko, A. G. Zelik, S. V. |
| author_facet | Ilyin, A. A. Kostianko, A. G. Zelik, S. V. |
| contents | 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 $Ω\subset \mathbb R^d$, $d\ge1$ with linear damping coefficient $γ>0$.
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 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 $d\ge2$ 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 $γ^{-1}$ in all spatial dimensions $d\ge1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18801 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp bounds on the attractor dimensions for damped wave equations Ilyin, A. A. Kostianko, A. G. Zelik, S. V. Analysis of PDEs 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 $Ω\subset \mathbb R^d$, $d\ge1$ with linear damping coefficient $γ>0$. 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 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 $d\ge2$ 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 $γ^{-1}$ in all spatial dimensions $d\ge1$. |
| title | Sharp bounds on the attractor dimensions for damped wave equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.18801 |