AI 中文总结
研究上半空间中三维等熵可压缩Navier-Stokes方程初边值问题，在滑移边界条件下，通过确定尺度不变初始量，在该量足够小的假设下，建立强解的整体适定性，给出柯西问题尺度不变整体理论的半空间对应物。
AI 中文摘要
我们研究了上半空间中具有滑移边界条件的三维等熵可压缩Navier-Stokes方程的初边值问题。我们证明了在存在真空和大振荡情况下强解的整体存在性和唯一性。虽然在几种情况下已经为有边界区域中的可压缩流开发了尺度框架，但上半空间中的整体适定性结果仍远未完善。具有远场真空的系统具有保持半空间几何形状和滑移边界条件的自然尺度结构。受此尺度的启发，我们确定了如下“尺度不变初始量”：$$ \left[ \|\rho_{0}\|_{L^{\infty}}^3 \left( \frac12\|\sqrt{\rho_0} u_0\|_{L^{2}}^{2} +\frac{1}{\gamma-1}\|P(\rho_{0})\|_{L^{1}} \right) +\|\rho_{0}\|_{L^{\infty}}^{\frac{3-\gamma}{2}} \right] \left( \|\nabla u_0\|_{L^2}^2 +\|P(\rho_{0})\|_{L^2}^2 \right). $$在该量足够小的假设下，我们建立了强解的整体适定性。该结果提供了由Wen（《高等数学》482（2025），论文编号110628）建立的柯西问题的尺度不变整体理论的半空间对应物，并表明滑移边界条件与该系统兼容。
英文摘要
We investigate the initial-boundary value problem for the three-dimensional isentropic compressible Navier--Stokes equations in the upper half-space with the slip boundary conditions. We prove the global existence and uniqueness of strong solutions in the presence of vacuum and large oscillations. Although scaling frameworks for compressible flows in domains with boundaries have been developed in several settings, the global well-posedness result in the half-space remains far from complete. The system with far-field vacuum admits a natural scaling structure that preserves both the half-space geometry and the slip boundary conditions. Motivated by this scaling, we identify the following \textit{scaling-invariant initial quantity}: $$ \left[ \|ρ_{0}\|_{L^{\infty}}^3 \left( \frac12\|\sqrt{ρ_0} u_0\|_{L^{2}}^{2} +\frac{1}{γ-1}\|P(ρ_{0})\|_{L^{1}} \right) +\|ρ_{0}\|_{L^{\infty}}^{\frac{3-γ}{2}} \right] \left( \|\nabla u_0\|_{L^2}^2 +\|P(ρ_{0})\|_{L^2}^2 \right). $$ Under the assumption that this quantity is sufficiently small, we establish the global well-posedness of strong solutions. This result provides a half-space counterpart of the scaling-invariant global theory for the Cauchy problem established by Wen (Adv. Math. 482 (2025), Paper No. 110628) and shows that the slip boundary condition is compatible with the system.
