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

In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces ${\dot{B}}_{p,r}^{\frac{d}{p}-1}$ 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 $\dot{\Delta}j u_0$\mathrm{.}Secondly,iftheinitialdata$u^n_0\rightarrow u_0$in$\dot{B}^{\frac{d}{p}-1}{p,r}$,thenthecorrespondinglifespan$T_n\rightarrow T$. Thirdly, we prove that the data-to-solutions map is continuous in ${\dot{B}}_{p,r}^{\frac{d}{p}-1}$. 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 $L^{\mathrm{\infty }}{L}^2\cap L^2{\dot{H}}^1$.