Energy Conservation for the Generalized Surface Quasi-geostrophic Equation

In this paper, we consider the generalized surface quasi-geostrophic equation with the velocity v determined by \(v=\mathcal {R}^{\perp }\Lambda ^{\gamma -1}\theta ,\) \(0<\gamma < 2\). It is shown that the \(L^p\)-norm of weak solutions is conserved provided \(\theta \in L^{p+1}\left( 0,T; {B}^{\frac{\gamma }{3}}_{p+1, c(\mathbb {N})}\right) \) for \(0<\gamma <\frac{3}{2}\) or \(\theta \in L^{p+1}\left( 0,T; {{B}}^{\alpha }_{p+1,\infty }\right) ~\text {for any}~\gamma -1<\alpha<1 \text { with} ~\frac{3}{2}\le \gamma <2\). Therefore, the accurate relationships between the critical regularity for the energy conservation of the weak solutions and the regularity of velocity for the generalized surface quasi-geostrophic equation are presented.