Continuum limits of Laplace operators on general lattices are considered, and it is shown that these operators converge to elliptic operators on the Euclidean space in the sense of the generalized norm resolvent convergence. We then study operators on the hexagonal lattice, which does not apply the above general theory, but we can show its Laplace operator converges to the continuous Laplace operator in the continuum limit. We also study discrete operators on the square lattice corresponding to second order strictly elliptic operators with variable coefficients, and prove the generalized norm resolvent convergence in the continuum limit. This talk is based on a joint work with Keita Mikami (RIKEN) and Shu Nakamura (Gakushuin University).

3次元全空間における定常外力付き圧縮性Navier-Stokes方程式の定常解の安定性について考察する。外力が小さいときの定常解の存在及び摂動問題の可解性についてはShibata-Tanaka(2003)による結果が既に得られている。本講演では定常解周りの摂動の正則性と時間大域的な挙動に関して得られた結果を述べる。またマッハ数と呼ばれる無次元数をゼロに近づけたときの特異極限問題について最近得られた結果を紹介する。