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 Ḃ d p −1 p,r with 1 ≤ r, p < ∞. We give a lower bound of the lifespan T which depends on the norm of the Littlewood-Paley decomposition of the initial data u0. Then we prove that if the initial data u n 0 → u0 in Ḃ d p −1 p,r , then the corresponding lifespan satisfies Tn → T , which implies that the common lower bound of the lifespan. Finally, we prove that the data-to-solutions map is continuous in Ḃ 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 [2, 7, 14], we deduce that Ḃ d p −1 p,∞ with 1 ≤ p < ∞ is the critical space which solutions are ill-posedness, while u ∈ Ḃ d p −1 p,r with 1 ≤ r, p < ∞ are well-poseness. Moreover, if we choose the initial data in a subset B̄ d p −1 p,∞ of Ḃ d p −1 p,∞ , we can obtain the well-posedness of the solutions.