Some remarks on a Hasegawa–Mima–Charney–Obukhov equation

Global Well-Posedness of an Inviscid Three-Dimensional Pseudo-Hasegawa-Mima Model

“…Global existence and uniqueness of solutions of the inviscid HMCO equation can be easily established following the same ideas as for the twodimensional Euler equations. Indeed, it was remarked in [10], and in [26], that (1.4), with F = β = 0, is equivalent to the two-dimensional Euler equations of incompressible inviscid fluid, where φ 0 plays the role of the stream function, ω = △ h φ 0 the vorticity and u = (− ∂φ0 ∂y , ∂φ0 ∂x ) is the velocity field. The existence and uniqueness of strong local solutions of (1.4) in H s (R 2 ), with initial data φ 0 ∈ H s (R 2 ), for s ≥ 4; and the existence of a weak global solution in H 2 (R 2 ), with initial data φ 0 ∈ H 2 (R 2 ), were established in [26].…”

“…Indeed, it was remarked in [10], and in [26], that (1.4), with F = β = 0, is equivalent to the two-dimensional Euler equations of incompressible inviscid fluid, where φ 0 plays the role of the stream function, ω = △ h φ 0 the vorticity and u = (− ∂φ0 ∂y , ∂φ0 ∂x ) is the velocity field. The existence and uniqueness of strong local solutions of (1.4) in H s (R 2 ), with initial data φ 0 ∈ H s (R 2 ), for s ≥ 4; and the existence of a weak global solution in H 2 (R 2 ), with initial data φ 0 ∈ H 2 (R 2 ), were established in [26]. The uniqueness of a global strong solution in H s (R 2 ), for s ≥ 4, was later established in [10].…”

Global strong solutions for the three-dimensional Hasegawa-Mima model with partial dissipation

“…Indeed, Guo and Han [7] proved the global existence and uniqueness of solutions for (1.6). For other results concerning (1.6) see, e.g., Paumond [19], and Gao and Zhu [6].…”

Global-in-time existence results for the two-dimensional Hasegawa–Wakatani equations

“…For the HM equation, we have had some mathematical results. For the initial value problem, the temporally local existence and uniqueness of the strong solution and the temporally global existence of the weak solution were proved by Guo and Han [13] and Paumond [40] independently in 2004, and the global existence of a strong solution was proved by Gao and Zhu [10] in 2005. The global-in-time existence and uniqueness of the solution and the existence of a global attractor to the initial boundary value problem for the generalized HM equation with periodic boundary condition were proved by Zhang and Guo for the twodimensional case [48] and the three-dimensional case [49].…”