The Continuous Dependence for the Navier–Stokes Equations in $\dot{B}^{\frac{d}{p}-1}_{p,r}$

Abstract

In this paper, we mainly investigate the Cauchy problem for the incompressible Navier–Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\le p<\infty ,\ 1\le r\le \infty , \ d\ge 2$. 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 $\dot{\Delta }_j u_0$. 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 is continuous in $\dot{B}^{\frac{d}{p}-1}_{p,r}$. 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 the well-posedness and weak-strong uniqueness results in $L^{\infty }L^2\cap L^{2}\dot{H}^1$.