The global attractors for the dissipative generalized Hasegawa-Mima equation (Q925970)

In this paper the following periodic initial value problem of the generalized Hasegawa-Mima equation with dissipative term \[ u_t-\Delta u_t- k_n u_y+ \gamma(u-\Delta u)+ f(u)_x+ h(u)= \{u,\Delta u\}+ g(x,y), \] \[ u(x,y,0)= u_0(x, y),\quad (x,y)\in \Omega= (0,2M)\times (0,2M), \] \[ u(x+ 2M, y+ 2M,t)= u(x,y,t),\quad t> 0, \] is considered. There \(\gamma\) and \(k_n\) are positive constants, \(\{f,g\}= (\partial_x f)(\partial_y g)- (\partial_y f)(\partial_x g)\), \(u(x,y, t)\) is an unknown real-valued function, \(f\), \(h\) and \(g\) are given real valued functions, \(f,h\in C^\infty\), \(g\in L^2(\Omega)\). First, the \(t\)-independent a priori estimates for the solutions are established. Then the existence of the global attractor of the periodic initial value problem is proved. Further, the estimates of the Hausdorff and fractal dimensions for the global attractor are given.