On the Global Well-posedness and Stability of the Navier—Stokes and the Related Equations

Abstract

We study the problem of global well-posedness and stability in the scale invariant Besov spaces for the modified 3D Navier-Stokes equations with the dissipation term, −Δu replaced by ${( - \Delta )^\alpha }u,0 \leqslant \alpha < \frac{5}{4}$. We prove the unique existence of a global-in-time solution in $B_{2,1}^{\frac{5}{2} - 2\alpha }$ for initial data having small $B_{2,1}^{\frac{5}{2} - 2\alpha }$ norm for all $\alpha \in \left[ {0,\frac{5}{4}} \right)$. We also obtain the global stability of the solutions $B_{2,1}^{\frac{5}{2} - 2\alpha }$ for $\alpha \in \left[ {\frac{1}{2},\frac{5}{4}} \right)$. In the case $\frac{1}{2} < \alpha < \frac{5}{4}$, we prove the unique existence of a global-in-time solution in $B_{p,\infty }^{\frac{3}{p} + 1 - 2\alpha }$ for small initial data, extending the previous results for the case α = 1.