Cauchy problem for the Navier-Stokes equation

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 equation.We first establish the local existence in the besov space with a lower bound of the lifespan which depends on the norm of the littlewood-paleydecomposition of the initial data.Then we prove that if the initial data in then the corresponding lifespan satisfies which implies that the common lower bound of the lifespan.Finally, we prove that the data-to-solutions map is continuous in so the solutions of navier-stokes equation are well-posedness (existence, uniqueness and continuous dependence) in the hadamard sense.