On the Serrin’s regularity criterion for the β-generalized dissipative surface quasi-geostrophic equation

Abstract

The authors establish a Serrin’s regularity criterion for the β-generalized dissipative surface quasi-geostrophic equation. More precisely, it is shown that if the smooth solution θ satisfies ∇θ ∈ L ^q(0, T;L ^q(R²)) with $\frac{\alpha }{q} + \frac{2}{p} \leqslant \alpha + \beta - 1$, then the solution θ can be smoothly extended after time T. In particular, when α + β ≥ 2, it is shown that if ∂yθ ∈ L ^q(0, T;L ^q(R²)) with $\frac{\alpha }{q} + \frac{2}{p} \leqslant \alpha + \beta - 1$, then the solution θ 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

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