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

TL;DR

This paper establishes local well-posedness and continuous dependence of solutions for the Navier-Stokes equations in critical homogeneous Besov spaces, with results on lifespan bounds and stability under initial data perturbations.

Contribution

It proves local existence, continuous dependence, and well-posedness of Navier-Stokes solutions in critical Besov spaces, extending understanding of solution behavior in these function spaces.

Findings

Solutions exist locally with a lifespan depending on initial data decomposition.

The lifespan of solutions is stable under initial data convergence.

The solution map is continuous in the Besov space topology.

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 $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. 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 well-posedness and weak-strong uniqueness results in $L^{\infty}L^2\cap L^{2}\dot{H}^1$.