Energy and helicity conservation for the generalized quasi-geostrophic equation

Abstract: In this paper, we consider the 2-D generalized surface quasi-geostrophic equation with the velocity $v$ determined by $v=\mathcal{R}^{\perp}\Lambda^{\gamma-1}\theta$. It is shown that the $L^p$ type energy norm of weak solutions is conserved provided $\theta\in L^{p+1}(0,T; {B}^{\frac{\gamma}{3}}_{p+1, c(\mathbb{N})})$ for $0<\gamma<\frac32$ or $\theta\in L^{p+1}(0,T; {{B}}^{\alpha}_{p+1,\infty})~\text{for any}~\gamma-1<\alpha<1 \text{ with} ~\frac{3}{2}\leq \gamma <2$. Moreover, we also prove that the helicity of weak solutions satisfying $\nabla\theta \in L^{3}(0,T;\dot{B}_{3,c(\mathbb{N})}^{\frac{\gamma}{3}})$ for $0<\gamma<\frac32$ or $\nabla\theta\in L^{3}(0,T; \dot{B}^{\alpha}_{3,\infty})~\text{for any}~\gamma-1<\alpha<1 \text{ with} ~\frac{3}{2}\leq \gamma <2$ 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.