Dirichlet Problem for Critical 2D Quasi-Geostrophic Equation with Large Data

J. Math. Sci. Univ. Tokyo
Vol. 28 (2021), No. 4, Page 557-582.

Dlotko, Tomasz; Liang, Tongtong; Wang, Yejuan
Dirichlet Problem for Critical 2D Quasi-Geostrophic Equation with Large Data
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Abstract:
The 2D Quasi-geostrophic equation attracts attention of mathematicians through recent years; see for example [2, 7, 8, 10, 12, 13, 16, 17, 20, 21, 39]. While the sub-critical problems are rather standard (but not classical), the critical equation contains nonlinearity of the same order as the main dissipative half negative Laplace operator. Therefore we face a balance of the two terms in that case, which makes the problem interesting. We construct a weak solution of the critical problem, and associate it with a multivalued semiflow, since the solution may not be unique. A compact global attractor is shown to exist for that multivalued semiflow.

Keywords: Quasi-geostrophic equation, fractional approximations, global attractor.

Mathematics Subject Classification (2010): 35S15, 35Q35, 35B41.
Received: 2020-03-30