Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations
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| Format: | Preprint |
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2024
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| author | Goldys, Beniamin Soenjaya, Agus L. Tran, Thanh |
| author_facet | Goldys, Beniamin Soenjaya, Agus L. Tran, Thanh |
| contents | We study the long-time behaviour of solutions to some classes of fourth-order nonlinear PDEs with non-monotone nonlinearities, which include the Landau--Lifshitz--Baryakhtar (LLBar) equation (with all relevant fields and spin torques) and the convective Cahn--Hilliard/Allen--Cahn (CH-AC) equation with a proliferation term, in dimensions $d=1,2,3$. Firstly, we show the global well-posedness, as well as the existence of global and exponential attractors with finite fractal dimensions for these problems. In the case of the exchange-dominated LLBar equation and the CH-AC equation without convection, an estimate for the rate of convergence of the solution to the corresponding stationary state is given. Finally, we show the existence of a robust family of exponential attractors when $d\leq 2$. As a corollary, exponential attractor of the LLBar equation is shown to converge to that of the Landau--Lifshitz--Bloch equation in the limit of vanishing exchange damping, while exponential attractor of the convective CH-AC equation is shown to converge to that of the convective Allen--Cahn equation in the limit of vanishing diffusion coefficient. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_09034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations Goldys, Beniamin Soenjaya, Agus L. Tran, Thanh Analysis of PDEs Dynamical Systems 35K52, 35Q60, 35B40, 37L30 We study the long-time behaviour of solutions to some classes of fourth-order nonlinear PDEs with non-monotone nonlinearities, which include the Landau--Lifshitz--Baryakhtar (LLBar) equation (with all relevant fields and spin torques) and the convective Cahn--Hilliard/Allen--Cahn (CH-AC) equation with a proliferation term, in dimensions $d=1,2,3$. Firstly, we show the global well-posedness, as well as the existence of global and exponential attractors with finite fractal dimensions for these problems. In the case of the exchange-dominated LLBar equation and the CH-AC equation without convection, an estimate for the rate of convergence of the solution to the corresponding stationary state is given. Finally, we show the existence of a robust family of exponential attractors when $d\leq 2$. As a corollary, exponential attractor of the LLBar equation is shown to converge to that of the Landau--Lifshitz--Bloch equation in the limit of vanishing exchange damping, while exponential attractor of the convective CH-AC equation is shown to converge to that of the convective Allen--Cahn equation in the limit of vanishing diffusion coefficient. |
| title | Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations |
| topic | Analysis of PDEs Dynamical Systems 35K52, 35Q60, 35B40, 37L30 |
| url | https://arxiv.org/abs/2411.09034 |