论文标题
$ \ dot {b}^{\ frac {d} {p} -1} _ {p，r} $中的navier-stokes方程的连续依赖
The continuous dependence for the Navier-Stokes equations in $\dot{B}^{\frac{d}{p}-1}_{p,r}$
论文作者
论文摘要
在本文中，我们主要调查了不可压缩的navier在同质besov空间中的方程$ \ dot {b}^{\ frac {\ frac {d} {p} {p} -1} -1} _ {p，r} $，带有$ 1 \ leq p <\ leq p <\\ infty，\ f \ iffty，\ fe fe fe f te uff inf inq flyq inf。首先，我们证明了解决方案的本地存在，并给出了解决方案的寿命$ t $的下限。寿命取决于初始数据的Littlewood-Paley分解，即$ \dotΔ_Ju_0 $。其次，如果初始数据$ u^n_0 \ rightarrow u_0 $ in $ \ dot {b}^{\ frac {d} {d} {p} {p} {p} -1} _ {p，r} $，则相应的lifespan $ t_n $ t_n \ rightarrow t $。第三，我们证明数据到溶液映射在$ \ dot {b}^{\ frac {d} {p} {p} -1} _ {p，r} $中是连续的。因此，在Hadamard意义上，Navier-Stokes方程的Cauchy问题在临界空间中是局部良好的。此外，我们还获得了良好的和虚弱的独特性，结果$ l^{\ infty} l^2 \ cap l^{2} \ dot {h}^1 $。
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 $\dotΔ_j u_0$. 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 $L^{\infty}L^2\cap L^{2}\dot{H}^1$.
