乘积型Sobolev不等式

时间：2014-02-25 19:51:13 阅读：252 评论：0 收藏：0 [点我收藏+]

(Multiplicative Sobolev inequality). Let $\mu,\lambda$ and $\gamma$ be three parameters that satisfy

1 \leq μ, λ < \infty, 2 λ + 1 μ > 1 and 1 + 3 γ = 2 λ + 1 μ .

$\bex 1\leq \mu,\lm<\infty,\quad \frac{2}{\lm}+\frac{1}{\mu}>1\quad\mbox{and}\quad 1+\frac{3}{\gamma}=\frac{2}{\lm}+\frac{1}{\mu}. \eex$ Assume

?∈H

)

$\phi\in H^1(\bbR^3)$ ,

$\p_1\phi$ ,

?∈L

)

$\p_2\phi\in L^\lm(\bbR^3)$ ,

?∈L

)

$\p_3\phi\in L^\mu(\bbR^3)$ . Then, there exists a constant

C=C(μ,λ)

$C=C(\mu,\lm)$ such that

∥ ? ∥ γ \leq C ∥ ? 1 ? ∥ 1 3 L λ ∥ ? 2 ? ∥ 1 3 L λ ∥ ? 3 ? ∥ 1 3 L μ .

$\bex \sen{\phi}_\gamma\leq C\sen{\p_1\phi}_{L^\lm}^\frac{1}{3} \sen{\p_2\phi}_{L^\lm}^\frac{1}{3} \sen{\p_3\phi}_{L^\mu}^{\frac{1}{3}}. \eex$

Reference:

C.S. Cao, J.H. Wu, Two regularity criteria for the $3$ D MHD equations, J. Differential Equations, 248 (2010), 2263--2274.

原文：http://www.cnblogs.com/zhangzujin/p/3566049.html

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