发表机构
Jiangxi University of Finance and Economics（江西财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明临界Besov空间中可压缩Navier-Stokes方程解在欧拉坐标下连续依赖于初值，无正则性损失，完善了Hadamard适定性。
AI 中文摘要
本文研究了临界Besov空间中重力可压缩Navier-Stokes方程解的连续依赖性（稳定性）。利用Danchin2014中发展的拉格朗日方法，证明了对于1<p<2d，流映射(a0,u0)↦(ā,ū)=(a∘X,u∘X)从Ḃ_{p,1}^{d/p}×Ḃ_{p,1}^{d/p-1}到C([0,T];Ḃ_{p,1}^{d/p})×E_p(T)是Lipschitz连续的。然而，由于临界初始数据的低正则性，这一结果并不能直接推出原始欧拉坐标中相应的连续依赖性。此前，仅对p<d，连续依赖性仅在相对于自然解空间损失一个导数的某些较低正则性空间中已知，这本质上是由唯一性论证的副产品。我们填补了这一空白，证明了对于1<p<2d，流映射(a0,u0)↦(a,u)在欧拉坐标中从Ḃ_{p,1}^{d/p}×Ḃ_{p,1}^{d/p-1}到C([0,T];Ḃ_{p,1}^{d/p})×E_p(T)是连续的（而非Lipschitz连续的），且没有任何正则性损失，这连同已知的存在唯一性理论，完善了临界空间中的Hadamard适定性。
英文摘要
This paper investigates the continuous dependence (stability) of solutions to the barotropic compressible Navier--Stokes equations in critical Besov spaces. Using the Lagrangian approach developed in \cite{Danchin2014}, it was shown that, for \(1<p<2d\), the flow map $(a_0,u_0)\mapsto (\bar a,\bar u)=(a\circ X,u\circ X)$ is Lipschitz continuous from $\dot B_{p,1}^{d/p}\times \dot B_{p,1}^{d/p-1}$ into $\mathcal{C}([0,T];\dot B_{p,1}^{d/p})\times E_p(T)$. However, this result does not directly imply the corresponding continuous dependence in the original Eulerian coordinates because of the low regularity of the critical initial data. Previously, only for $p<d$, continuous dependence was known only in certain lower-regularity spaces with a loss of one derivative relative to the natural solution spaces arising essentially as a by-product of the uniqueness argument. We close this gap and prove that for $1<p<2d$ the flow map $(a_0,u_0)\mapsto(a,u)$ is continuous (not Lipschitz continuous) from $\dot B_{p,1}^{d/p}\times\dot B_{p,1}^{d/p-1}$ to $\mathcal{C}([0,T];\dot B_{p,1}^{d/p})\times E_p(T)$ in Eulerian coordinates without any loss of regularity, which together with the known existence and uniqueness theory \cite{Danchin2014} completes Hadamard well-posedness in the critical spaces.
