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In this paper, we mainly investigate the Cauchy problem for theincompressible 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 alower bound of the lifespan $T$ of the solution. The lifespan depends on theLittlewood-Paley decomposition of the initial data, that is $\dot{\Delta}_ju_0$. 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 iscontinuous in $\dot{B}^{\frac{d}{p}-1}_{p,r}$. Therefore, the Cauchy problem ofthe Navier-Stokes equations is locally well-posed in the critical Besov spacesin the Hadamard sense. Moreover, we also obtain well-posedness and weak-stronguniqueness results in $L^{\infty}L^2\cap L^{2}\dot{H}^1$.

The continuous dependence for the Navier-Stokes equations in $\dot{B}^{\frac{d}{p}-1}_{p,r}$