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In this paper, we first write the velocity equation of the Bénard system in its two components, and consider the global weak solution of the resulting 2D Bénard system with partial dissipation, i.e. (1) \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (2) \(\mu _{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (3) \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa _{1}>0\), \(\kappa_{2}=0\); (4) \(\mu_{1}>0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (5) \(\mu_{1}=0\), \(\mu_{2}>0\), \(\mu _{3}=0\), \(\mu_{4}=0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\); (6) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}>0\), \(\mu_{4}=0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (7) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu_{4}>0\), \(\kappa_{1}>0\), \(\kappa_{2}=0\); (8) \(\mu_{1}=0\), \(\mu_{2}=0\), \(\mu_{3}=0\), \(\mu _{4}>0\), \(\kappa_{1}=0\), \(\kappa_{2}>0\), where \(\mu_{i}\) (\(i=1,2,3,4\)) and \(\kappa_{j}\) (\(j=1,2\)) are the coefficients of dissipation and thermal diffusivity. Furthermore, we establish some regularity criteria for the corresponding system. This work follows the techniques in the paper of Cao and Wu (Adv. Math. 226:1803–1822, 2011).

Global existence of weak solution and regularity criteria for the 2D Bénard system with partial dissipation