Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces

• Published in: SIAM journal on mathematical analysis 2005, Vol.36 (3), p.1014-1030
• Main Author: Wu, Jiahong
• Format: Article
• Language: English
• Subjects: Computational methods in fluid dynamics; Estimates; Exact sciences and technology; Fluid dynamics; Fourier transforms; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematical problems; Mathematics; Navier-Stokes equations; Partial differential equations; Physics; Rotational flow and vorticity; Sciences and techniques of general use
• ISSN: 0036-1410; 1095-7154
• DOI: 10.1137/s0036141003435576

The two-dimensional (2D) quasi-geostrophic (QG) equation is a 2D model of the 3D incompressible Euler equations, and its dissipative version includes an extra term bearing the operator $(-\Delta)^\alpha$ with $\alpha\in [0,1]$. Existing research appears to indicate the criticality of $\alpha=\frac12$ in the sense that the issue of global existence for the 2D dissipative QG equation becomes extremely difficult when $\alpha\le \frac12$. It is shown here that for any $\alpha\le \frac12$ the 2D dissipative QG equation with an initial datum in the Besov space $B^r_{2,\infty}$ or $B^r_{p,\infty}$ $(p>2)$ possesses a unique global solution if the norm of the datum in these spaces is comparable to $\kappa$, the diffusion coefficient. Since the Sobolev space $H^r$ is embedded in $B^r_{2,\infty}$, a special consequence is the global existence of small data solutions in $H^r$ for any $r>2-2\alpha$.