The PDEs and Applications seminar: Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

  • Date: –12:00
  • Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
  • Lecturer: Anton Savostianov
  • Organiser: Matematiska institutionen
  • Contact person: Kaj Nyström
  • Seminarium

Abstract: It is well known that long time behaviour of  a dissipative dynamical system generated by an evolutionary PDE can be described in terms of attractor, an attracting set which is essentially thinner than a ball of the corresponding phase space of the system. In this talk we compare long time behaviour of damped anisotropic wave equations with the corresponding homogenised limit in terms of their attractors. First we will formulate order sharp estimates between the trajectories of the corresponding systems and will see that the hyperbolic nature of the problem results in extra correction comparing with parabolic equations. Then, after brief review on previous results on homogenisation of attractors, we will discuss new results. It appears that the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts, in suitable norms, can be estimated via operator norm of the difference of the resolvents of the corresponding elliptic operators. Furthermore, we show that the homogenised attractor admits first-order correction suggested by the natural asymptotic expansion. The corrected homogenised attractors, as expected, are close to the anisotropic attractors already in the strong energy norm. The corresponding quantitative estimates on the Hausdorff distance between the corrected homogenised attractors and anisotropic ones, with respect to the strong energy norm, are also obtained. Our results are applied to Dirchlet, Neumann and periodic boundary conditions. This is joint work with Shane Cooper.