On dimension of the global attractor for 2D quasi-geostrophic equations

We obtain a precise upper bound of the fractal dimension of the global attractor for 2D quasi-geostrophic equations. The upper bound is a decreasing function of the coefficient κκ of dissipative term, which conforms to physical intuition. Moreover, the bound tends to infinity as κ→0κ→0 and α→12, which reflects the chaotic behavior of the QG equation without dissipative effect and in critical case, respectively. Our result gives an answer to a problem posed in Ju [N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Commun. Math. Phys., 255 (2005) 161–181].