On the global solvability of the generalised Navier-Stokes system in critical Besov spaces

This paper is devoted to the global solvability of the Navier-Stokes system with fractional Laplacian $(-\mathrm{\Delta }{)}^{\alpha }$ in ${\mathbb{R}}^n$ for $n\ge 2$, where the convective term has the form $(\mathrm{\mid }u{\mathrm{\mid }}^{m-1}u)\cdot \mathrm{\nabla }u$ for $m\ge 1$. By establishing the estimates for the difference $\mathrm{\mid }{u}_1{\mathrm{\mid }}^{m-1}{u}_1-\mathrm{\mid }{u}_2{\mathrm{\mid }}^{m-1}{u}_2$ in homogeneous Besov spaces, and employing the maximal regularity property of $(-\mathrm{\Delta }{)}^{\alpha }$ in Lorentz spaces, we prove global existence and uniqueness of the strong solution of the Navier-Stokes in critical Besov spaces for both $m=1$ and $m>1$