Global solution of the 3‐D incompressible Navier‐Stokes equations in the Besov spaces B˙r1,r2,r3σ,1

In this paper, we construct a more general Besov spaces ${\dot{B}}_{r_1,r_2,r_3}^{\sigma ,q}$ and consider the global well‐posedness of incompressible Navier‐Stokes equations with small data in ${\dot{B}}_{r_1,r_2,r_3}^{\sigma ,\infty }$ for $\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}?\sigma =1,1?r_i<\infty$. In particular, we show that for any $\frac{2}{\gamma }+\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}?s=1,1<\gamma <\infty$ and $r_i?p_i<\infty$, the solution with initial data in ${\dot{B}}_{r_1,r_2,r_3}^{\sigma ,\infty }$ belong to , which, as far as we know, has not been discussed in other papers. Moreover, the smoothing effect of the solution to Navier‐Stokes equations is proved, which may have its own interest.