AI 中文总结
本文研究带势外力的三维可压缩Navier–Stokes方程柯西问题，证明对任意大L²初值的整体适定性，给出最优时间衰减率，分析依赖齐次能量估计与频率局域化方法。
AI 中文摘要
我们研究三维等熵可压缩Navier–Stokes方程的柯西问题，该方程带有不随时间变化的势外力，且在空间非恒定稳态附近展开。势函数由无权重齐次Besov空间控制，特别地，不施加涉及(1+|x|)^j∇^jφ的多项式空间权重条件。对于相对于稳态的初值，若其在$\dot H^{\frac12-\delta}\cap\dot H^3$中足够小，我们建立了$H^3$中整体强解的存在性与唯一性，同时允许初始$L^2$范数任意大。若初值在$\dot B^s_{2,\infty}$中有界，其中$s\in[-\frac32,-1)$，则解及其一阶空间导数分别以最优速率$(1+t)^{-\frac{k-s}{2}}$衰减，其中$k=0$和$1$。该分析依赖于精细的齐次能量估计以及对系统耗散结构与渐近结构的频率局域化描述。
英文摘要
We study the Cauchy problem for the three-dimensional barotropic compressible Navier--Stokes equations with a time-independent potential force near a spatially nonconstant stationary state. The potential is controlled in unweighted homogeneous Besov spaces; in particular, no polynomial spatial-weight condition involving $(1+|x|)^j\nabla^jϕ$ is imposed. For initial data relative to the stationary state that are sufficiently small in $\dot H^{\frac12-δ}\cap\dot H^3$, we establish the existence and uniqueness of a global strong solution in $H^3$, while allowing the initial $L^2$ norm to be arbitrarily large. If the initial data are bounded in $\dot B^s_{2,\infty}$ for $s\in[-\frac32,-1)$, then the solution and its first spatial derivative decay at the optimal rates $(1+t)^{-\frac{k-s}{2}}$ with $k=0$ and $1$, respectively. The analysis relies on refined homogeneous energy estimates and a frequency-localized description for the dissipative and asymptotic structures of the system.
Comments36 pages. All comments are welcome
