论文标题
通用准斑块方程的能量和螺旋性保护
Energy and helicity conservation for the generalized quasi-geostrophic equation
论文作者
论文摘要
在本文中，我们考虑了2-D广义的表面准斑块方程，其速度$ V $由$ v = \ Mathcal {r}^{\ perp}λ^{γ-1}θ$确定。结果表明，$ l^p $类型的弱解决方案的能量规范保存在l^{p+1}中提供$θ\（0，t; {b}^{\fracγ{3}} _ {p+1，c（\ mathbb {n}）} $ $ 0 <frac32 $或l^{p+1}（0，t; {b}^α_{p+1，\ infty}）此外，我们还证明了满足$ \nablaθ\ in l^{3}（0，t; \ dot {b} _ {3，c（\ mathbb {n}）}^{\fracγ{3}} $ 0 <grac的uca的弱点的螺旋性。 l^{3}（0，t; \ dot {b}^α_{3，\ infty}）因此，提出了弱解的临界（螺旋）保守性的临界规则性与二维广义准整形方程中速度的规则性之间的准确关系。
In this paper, we consider the 2-D generalized surface quasi-geostrophic equation with the velocity $v$ determined by $v=\mathcal{R}^{\perp}Λ^{γ-1}θ$. It is shown that the $L^p$ type energy norm of weak solutions is conserved provided $θ\in L^{p+1}(0,T; {B}^{\fracγ{3}}_{p+1, c(\mathbb{N})})$ for $0<γ<\frac32$ or $θ\in L^{p+1}(0,T; {B}^α_{p+1,\infty})~\text{for any}~γ-1<α<1 \text{ with} ~\frac{3}{2}\leq γ<2$. Moreover, we also prove that the helicity of weak solutions satisfying $\nablaθ\in L^{3}(0,T;\dot{B}_{3,c(\mathbb{N})}^{\fracγ{3}})$ for $0<γ<\frac32$ or $\nablaθ\in L^{3}(0,T; \dot{B}^α_{3,\infty})~\text{for any}~γ-1<α<1 \text{ with} ~\frac{3}{2}\leq γ<2$ 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.
