AI 中文总结
本文证明三维可压缩Navier-Stokes方程半空间外流问题中弱粘性激波在小扰动下全局渐近稳定，结合a-收缩法与高阶能量估计，首次给出多维外流问题的激波稳定性结果。
AI 中文摘要
我们建立了半空间中三维正压可压缩Navier-Stokes方程外流问题的平面粘性激波的渐近稳定性，其中在横向方向上施加周期边界条件。对于距离边界足够远的弱激波，我们证明，在$H^{2}$中的小扰动下，外流问题存在唯一的全局时间解，并且该解一致收敛到粘性激波（允许一个动力学位移），其速度随时间渐近衰减。这为多维外流问题提供了首个激波稳定性结果。我们的证明结合了$a$-收缩方法和适用于半空间边界的高阶能量估计。零阶估计中剩余的边界迹线通过沿切向方向对扰动系统求导并利用外流通量的有利符号来控制。然后通过动量方程从时间和切向导数恢复最高阶法向导数，而由动力学位移产生的边界项则利用粘性激波尾部的指数衰减来控制。
英文摘要
We establish the asymptotic stability of planar viscous shock waves for the outflow problem of the three-dimensional barotropic compressible Navier--Stokes equations in a half-space, with periodic boundary conditions imposed in the transverse directions. For a weak shock located sufficiently far from the boundary, we prove that, under small perturbations in $H^{2}$, the outflow problem admits a unique global-in-time solution and that the solution converges uniformly to the viscous shock, up to a dynamical shift, whose velocity time-asymptotically decays. This provides the first shock-stability result for the multidimensional outflow problem. Our proof combines the $a$-contraction method and higher-order energy estimates adapted to the half-space boundary. The boundary trace remaining in the zeroth-order estimate is controlled by differentiating the perturbation system in the tangential directions and exploiting the favorable sign of the outflow flux. The highest-order normal derivatives are then recovered from the time and tangential derivatives through the momentum equation, while the boundary terms generated by the dynamical shift are controlled using the exponential decay of the viscous shock tail.
Comments47 pages
