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

Abstract

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 $1\le p<\infty ,1\le r\le \infty ,d\ge 2$. 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 $u_0^n\to u_0$ in ${\dot{B}}_{p,r}^{\frac{d}{p}-1}$, then the corresponding lifespan $T_n\to T$. 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 $L^{\infty }{L}^2\cap L^2{\dot{H}}^1$.

arXiv admin note: text overlap with arXiv:2012.03489