About lifespan and the continuous dependence for the Navier-Stokes equation in $\dot{B}^{\frac{d}{p}-1}_{p,r}$

In this paper, we mainly investigate the Cauchy problem for the Navier-Stokes ($NS$) equation. We first establish the local existence in the Besov space $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq r,p<\infty$. We give a lower bound of the lifespan $T$ which depends on the norm of the Littlewood-Paley decomposition of the initial data $u_0$. Then we prove that if the initial data $u^n_0\rightarrow u_0$ in $\dot{B}^{\frac{d}{p}-1}_{p,r}$, then the corresponding lifespan satisfies $T_n\rightarrow T$, which implies that the common lower bound of the lifespan. Finally, we prove that the data-to-solutions map is continuous in $\dot{B}^{\frac{d}{p}-1}_{p,r}$. So the solutions of Navier-Stokes equation are well-posedness (existence, uniqueness and continuous dependence) in the Hadamard sense. Combining \cite{wbx,ns7,ns2}, we deduce that $\dot{B}^{\frac{d}{p}-1}_{p,\infty}$ with $1\leq p<\infty$ is the critical space which solutions are ill-posedness, while $u\in\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq r,p<\infty$ are well-poseness. Moreover, if we choose the initial data in a subset $\bar{B}^{\frac{d}{p}-1}_{p,\infty}$ of $\dot{B}^{\frac{d}{p}-1}_{p,\infty}$, we can obtain the well-posedness of the solutions.

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