Energy and helicity conservation for the generalized quasi-geostrophic equation

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