圆柱形区域中粘性可压缩流体steady flow的非齐次滑移-流入边值问题
On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain
摘要：
本文研究了圆柱形区域$Ω= Ω_0 \times (0,L) \in \mathbb{R}^3$中粘性可压缩流体的定常流动，其中密度为流入边界条件，速度为非均匀滑移边界条件。我们表明存在一个解决方案$(v,ρ) \in W^2_p(Ω) \times W^1_p(Ω)$，其中$v$是流体的速度和$ρ$是密度，这是一个小扰动的恒定流$(\bar v \equiv [1,0,0], \bar ρ\equiv 1)$。我们还证明了该解在$(\bar v,\bar ρ)$的一类小扰动下是唯一的。连续性方程中的项$u \cdot \nabla w$使得直接应用不动点方法不可能证明存在性。因此，为了证明解的存在性，我们构造了一个序列$(v^n,ρ^n)$，该序列在$W^2_p(Ω) \times W^1_p(Ω)$中有界，并且在更大的空间$L_{\infty}(0,L;L_2(Ω_0))$中满足Cauchy条件，这使得我们能够推导出$(v^n,ρ^n)$的子序列的弱极限实际上是我们问题的强解。
英文摘要：
We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain $Ω= Ω_0 \times (0,L) \in \mathbb{R}^3$. We show existence of a solution $(v,ρ) \in W^2_p(Ω) \times W^1_p(Ω)$, where $v$ is the velocity of the fluid and $ρ$ is the density, that is a small perturbation of a constant flow $(\bar v \equiv [1,0,0], \bar ρ\equiv 1)$. We also show that this solution is unique in a class of small perturbations of $(\bar v,\bar ρ)$. The term $u \cdot \nabla w$ in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence $(v^n,ρ^n)$ that is bounded in $W^2_p(Ω) \times W^1_p(Ω)$ and satisfies the Cauchy condition in a larger space $L_{\infty}(0,L;L_2(Ω_0))$ what enables us to deduce that the weak limit of a subsequence of $(v^n,ρ^n)$ is in fact a strong solution to our problem.
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