Energy and helicity conservation for the generalized quasi-geostrophic equation

TL;DR

This paper investigates conditions under which energy and helicity are conserved in weak solutions of the 2-D generalized quasi-geostrophic equation, linking regularity of solutions to conservation laws.

Contribution

It establishes precise regularity criteria for energy and helicity conservation in weak solutions of the generalized quasi-geostrophic equation.

Findings

Energy norm conservation under specific Besov space conditions.

Helicity invariance for solutions with certain gradient regularity.

Relationship between solution regularity and conservation laws.

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.