Global Well-Posedness and Asymptotic Behavior for the 2D Subcritical Dissipative Quasi-Geostrophic Equation in Critical Fourier-Besov-Morrey Spaces (Journal of Partial Differential Equations, 2023)
In this paper, we study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists $v$ a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces $ \mathcal{F} {\mathcal{N}}_{p, λ, q}^{3-2 α+\frac{λ-2}{p}}$. Moreover, we show the asymptotic behavior of the global solution $v$. i.e., $\|v(t)\|_{ \mathcal{F} {\mathcal{N}}_{p, λ, q}^{3-2 α+\frac{λ-2}{p}}}$ decays to zero as time goes to infinity.
Keywords: 2D quasi-geostrophic equation; subcritical dissipation; Littlewood-Paley theory; global well-posedness; long time behavior of the solution; Fourier-Besov-Morrey spaces.