  In this paper, we mainly investigate the Cauchy problem for the
incompressible Navier-Stokes equations in homogeneous Besov spaces
$\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \
d\geq 2$. Firstly, we prove the local existence of the solution and give a
lower bound of the lifespan $T$ of the solution. The lifespan depends on the
Littlewood-Paley decomposition of the initial data, that is $\dot{\Delta}_j
u_0$. Secondly, if the initial data $u^n_0\rightarrow u_0$ in
$\dot{B}^{\frac{d}{p}-1}_{p,r}$, then the corresponding lifespan
$T_n\rightarrow T$. Thirdly, we prove that the data-to-solutions map is
continuous in $\dot{B}^{\frac{d}{p}-1}_{p,r}$. Therefore, the Cauchy problem of
the Navier-Stokes equations is locally well-posed in the critical Besov spaces
in the Hadamard sense. Moreover, we also obtain well-posedness and weak-strong
uniqueness results in $L^{\infty}L^2\cap L^{2}\dot{H}^1$.
