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Long-time behavior for 2D non-autonomous $g$-Navier–Stokes equations

Volume 103 / 2012

Cung The Anh, Dao Trong Quyet Annales Polonici Mathematici 103 (2012), 277-302 MSC: Primary 35B41; Secondary 35Q30, 37L30, 35D05. DOI: 10.4064/ap103-3-5

Abstract

We study the first initial boundary value problem for the 2D non-autonomous $g$-Navier–Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite-dimensional pullback $\mathcal D_\sigma$-attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Furthermore, when the force is time-independent and “small”, the existence, uniqueness and global stability of a stationary solution are also studied.

Authors

  • Cung The AnhDepartment of Mathematics
    Hanoi National University of Education
    136 Xuan Thuy, Cau Giay
    Hanoi, Vietnam
    e-mail
  • Dao Trong QuyetFaculty of Information Technology
    Le Qui Don Technical University
    100 Hoang Quoc Viet, Cau Giay
    Hanoi, Vietnam
    e-mail

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