Homogenization of Stokes with Neumann condition on oscillating boundary

Article type: Research Article

Authors: Muthukumar, Thirumalai Nambi | Sardar, Bidhan Chandra^{; *}

Affiliations: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttarpradesh-208016, India. E-mails: tmk@iitk.ac.in, bcsardar@iitk.ac.in

Correspondence: [*] Corresponding author. E-mail: bcsardar@iitk.ac.in.

Abstract: The stationary Stokes problem in a n-dimensional domain with a rapidly oscillating (n−1) dimensional boundary prescribed with Neumann boundary condition and periodicity along the lateral sides is considered. We identify the limit equation in a fixed domain with a one dimensional Laplacian-type problem, satisfied on the region covered by the oscillating part of the boundary, coupled with steady Stokes in the complement. The existence and uniqueness of solution for the coupled limit problem is established and, finally, the weak convergences of velocities are improved to strong convergence by introducing corrector terms.

Keywords: Rough boundary, unfolding operator, boundary homogenization

DOI: 10.3233/ASY-181499

Journal: Asymptotic Analysis, vol. 112, no. 3-4, pp. 125-150, 2019