2021/04/28 by Yanan Li, Zhijian Yang, Li, Yanan +1
Engineering · Mathematics · Computer Science · #Stability and Controllability of Differential Equations #Navier-Stokes equation solutions #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2104.13606
The paper is concerned with the exponential attractors for the viscoelastic wave model in Ω⊂ \mathbb R3: utt-ht(0)Δu-∫0^∞∂sht(s)Δu(t-s)\mathrm ds+f(u)=h, with time-dependent memory kernel ht(⋅) which is used to model aging phenomena of the material. Conti et al [Amer. J. Math., 2018] recently provided the correct mathematical setting for the model and a well-posedness result within the novel theory of dynamical systems acting on. time-dependent spaces, recently established by Conti, Pata and Temam [J. Differential Equations, 2013], and proved the existence and the regularity of the time-dependent global attractor. In this work, we further study the existence of the time-dependent exponential attractors as well as their regularity. We establish an abstract existence criterion via quasi-stability method introduced originally by Chueshov and Lasiecka [J. Dynam. Diff.Eqs.,2004], and on the basis of the theory and technique developed in [Amer. J. Math., 2018] we further provide a new method to overcome the difficulty of the lack of further regularity to show the existence of the time-dependent exponential attractor. And these techniques can be used to tackle other hyperbolic models.