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 $v={\mathcal{R}}^{\perp }\hat{I}{›}^{{\hat{I}}^3-1}\hat{I}¸$. It is shown that the $L^p$ type energy norm of weak solutions is conserved provided $\hat{I}¸\in L^{p+1}(0,T;B_{p+1,c(\mathbb{N})}^{\frac{\hat{I}}{³}3})$ for $0<{\hat{I}}^3<\frac{3}{2}$ or $\hat{I}¸\in L^{p+1}(0,T;B^{\hat{I}}{\pm }_{p+1,\infty })\text{for any}{\hat{I}}^3-1<\hat{I}\pm <1\text{ with}\frac{3}{2}\le {\hat{I}}^3<2$. Moreover, we also prove that the helicity of weak solutions satisfying $\nabla \hat{I}¸\in L^3(0,T;{\dot{B}}_{3,c(\mathbb{N})}^{\frac{\hat{I}}{³}3})$ for $0<{\hat{I}}^3<\frac{3}{2}$ or $\nabla \hat{I}¸\in L^3(0,T;{\dot{B}}^{\hat{I}}{\pm }_{3,\infty })\text{for any}{\hat{I}}^3-1<\hat{I}\pm <1\text{ with}\frac{3}{2}\le {\hat{I}}^3<2$ 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.

21 pages