Abstract
This paper is devoted to the study of the modified quasi-geostrophic equation \begin{document}$ \partial_t\theta+u\cdot\nabla\theta+\nu\Lambda^\alpha\theta = 0 \ \ \mbox{ with } \ \ u = \Lambda^\beta\mathcal{R}^\perp\theta $\end{document} in $ \mathbb{R}^2 $. By the Littlewood-Paley theory, we obtain the local well-posedness and the smoothing effect of the equation in critical Besov spaces. These results are applied to show the global existence of regular solutions for the critical case $ \beta = \alpha-1 $ and the existence of regular solutions for large time $ t>T $ with respect to the supercritical case $ \beta >\alpha -1 $ in Besov spaces. Earlier results for the equation in Hilbert spaces $ H^s $ spaces are improved.
Highlights
Consider the following modified quasi-geostrophic equation [21]∂tθ + u · ∇θ + νΛαθ = 0, (1) u = ΛβR⊥θ, θ|t=0 = θ0 in R2 for the parameters ν > 0, α ∈ (0, 2) and β ∈ (0, 1) and the operators Λα = (−∆) α2 and R⊥ = (−∂x2 Λ−1, ∂x1 Λ−1)
The analysis on (1) in Besov spaces is essentially based on the invariance property of (1) with respect to the scaling transformations θλ = λα−β−1θ(λx, λαt) and uλ = λα−1θ(λx, λαt)
The purpose of the present paper is to investigate the existence of regular solutions to (1) in critical homogeneous Besov spaces with respect to the invariance of (1) under the scaling transformation (2)
Summary
In order to use the second step approach, they assumed u ∈ L∞,loc(0, ∞; C1−α(R2)), and proved that the Leray-Hopf weak solution θ. Almost at the same time, Constantin and Wu [11] proved that for δ > 1 − α, the Leray-Hopf weak solution θ of (3) with α < 1 belongs to C∞((t0, t] × R2) if θ ∈ L∞([t0, t]; Cδ(R2)) This result was later improved by Dong and Pavlovic [18, 19] for the case δ 1 − α. The velocity u = Λα−1R⊥θ ∈ L∞(0, ∞; C1−α(R2)) for the Leray-Hopf weak solution θ ∈ L∞(R2 × (0, ∞)) obtained by following the first step examination of [3]
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