The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case
Discrete and continuous dynamical systems. Series B
We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature. 2010 Mathematics Subject Classification. Primary: 35Q86, 35R11; Secondary: 34D45. 2531 2532 WENRU HUO AND AIMIN HUANG Moreover, integrating (1) on Ω and integration by parts yield
... ,f are the mean of u, θ, f over Ω respectively; that is Therefore, with loss of generality, we assume that u, θ, f are all of mean zero. Otherwise, we can replace u −ū, θ −θ, f −f by u, θ, f respectively. Recently, the 2D Boussinesq equations and their fractional generalizations have attracted considerable attention due to their physical applications and mathematical significance. When α = β = 1, the system (1) is then called the standard 2D Boussinesq equations, which are widely used to model the geophysical flows such as atmospheric fronts and oceanic circulation and also play an important role in the study of Rayleigh-Bénard convection (c.f.  ). Besides, the 2D Boussinesq system are the two-dimensional models which retain the key vortex-stretching mechanism as the 3D Navier-Stokes/Euler equations for axisymmetric swirling flows (c.f.  ). In the physical aspects, although the incompressibility and Boussinesq approximations are applicable, flows in the middle atmosphere traveling upwards change because of the changes of atmospheric properties. The effect of kinematic viscosity and thermal diffusion becomes attenuated since the thinning of atmosphere. This anomalous attenuation phenomenon can be modeled by using the fractional Laplacian. In addition, some related models with fractional Laplacian such as the surface quasi-geostrophic equation indeed are from geophysics and are very significant in physical applications. In the mathematical respect, the global well-posedness, global regularity of the standard 2D Boussinesq system as well as the existence of the global attractor have been widely studied, see for example [3, 9, 8, 6, 31, 38, 5, 39, 23] . Recently, there are many works devoted to the study of the 2D Boussinesq system with partial viscosity, see for example [17, 4, 13, 7, 14, 15] in the whole space R 2 and [44, 29, 16, 19] in bounded domains. There are also many works which considered the global regularity of 2D Boussinesq system with fractional diffusion, see for example [41, 20, 42, 43, 34] . The Boussinesq systems with fractional Laplacian pose more of a challenge than that with the full Laplacian and allow us to study a family of equations simultaneously. In some realistic applications, the variation of the viscosity and diffusivity with the temperature may not be disregarded (see for example  and references therein) and there are many works on this direction, too, see for example [25, 26, 35, 28, 18] where the existence of weak solutions, global regularity, and existence of global attractor have been studied. However, the global regularity for the inviscid 2D Boussinesq system where ν = κ = 0 is still an outstanding open problem and the study of fractional Boussinesq system may shed some light on the inviscid Boussinesq system. To the best of our knowledge, the existence of the global attractor for the 2D Boussinesq equations with fractional dissipation has not been addressed yet, which is the goal of this article. This work is motivated by the [2, 21] , where the existence of global attractor of the 2D subcritical SQG equations has been proved. The key point in [2, 21] is that they proved a positivity lemma for the fractional Laplacian. Armed with the positivity lemma, we can similarly obtain the maximum principle for θ as in [2, 21] and then follow the standard procedure to show the existence of global attractor.