Valery Smyshlyaev (UCL)

Two-scale homogenisation for a general class of high-contrast PDE systems

Abstract:
I present recent work with Ilia Kamotski [1] on two-scale homogenisation of general PDE systems with periodic coefficients with a critically scaled high contrast, which reflects certain interesting effects due to underlying ``micro-resonances’’. It appears that a strong two-scale resolvent convergence of associated high-contrast elliptic operators holds under a rather generic decomposition assumption. This implies in particular (two-scale) convergence of parabolic and hyperbolic semigroups with applications to a wide class of initial value problems. In the end I briefly discuss most recent stronger results with operator-type error estimates for high-contrast problems (with Shane Cooper and I. Kamotski), as well as situations where the micro-resonances display certain randomness. In simplest cases, the resulting two-scale limit behaviour appears to be rather explicit and the macroscopic equations display a form of wave trapping by the micro-resonances due to their randomness.

[1] I.V. Kamotski, V.P. Smyshlyaev, Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients, {\it Applicable Analysis} 98 (1-2), 64--90 (2019).