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| On the Serrin's Regularity Criterion for the β-Generalized Dissipative Surface Quasi-geostrophic Equation |
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Citation： |
Jihong ZHAO,Qiao LIU.On the Serrin's Regularity Criterion for the β-Generalized Dissipative Surface Quasi-geostrophic Equation[J].Chinese Annals of Mathematics B,2015,36(6):947~956 |
| Page view： 1189
Net amount： 953 |
Authors： |
Jihong ZHAO; Qiao LIU |
Foundation： |
This work was supported by
the National Natural Science Foundation of China (Nos.\,11501453, 11371294, 11326155,
11401202), the Fundamental Research Funds for the Central
Universities (No.\,2014YB031), the Fundamental Research Project of
Natural Science in Shaanxi Province--Young Talent Project
(No.\,2015JQ1004), the Scientific Research Fund of Hunan Provincial
Education Department (No.\,14B117) and the China Postdoctoral
Science Foundation (No.\,2015M570053). |
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| Abstract： |
The authors establish a Serrin's regularity criterion for the
$\beta$-generalized dissipative surface quasi-geostrophic equation.
More precisely, it is shown that if the smooth solution $\theta$
satisfies $\nabla \theta\in L^{q}(0,T; L^{p}(\mathbb{R}^{2}))$ with
$\frac{\alpha}{q}+\frac{2}{p}\leq\alpha+\beta-1$, then the solution
$\theta$ can be smoothly extended after time $T$. In particular,
when $\alpha+\beta\geq2$, it is shown that if $\partial_{y}
\theta\in L^{q}(0,T; L^{p}(\mathbb{R}^{2}))$ with
$\frac{\alpha}{q}+\frac{2}{p}\leq\alpha+\beta-1$, then the solution
$\theta$ can also be smoothly extended after time $T$. This result
extends the regularity result of Yamazaki in 2012. |
Keywords： |
β-generalized quasi-geostrophic equation, Weak solution,
Serrin's regularity criterion |
Classification： |
35B65, 35Q86, 35R11 |
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