正压 Navier-Stokes 方程不透墙问题与入流问题向粘性激波的渐近行为
Asymptotic behavior toward viscous shock for impermeable wall and inflow problem of barotropic Navier-Stokes equations
AI总结：
本文研究半直线上可压缩正压 Navier-Stokes 方程不透墙与入流问题向传出粘性激波的渐近收敛，利用带平移的 $a$-收缩方法，在小 $H^1$ 初始扰动下证明一致收敛并放宽了既有变量与 $\gamma$ 假设。
摘要：
我们考虑半直线上的可压缩正压 Navier-Stokes 方程，并研究其向传出粘性激波的时间渐近行为。具体而言，我们考虑两个边界问题：不透墙问题和入流问题，其中边界处的速度被给定为常数状态。对于这两个问题，当由边界和远场所给定的常数状态确定的渐近剖面是粘性激波时，我们证明：在初始扰动在 $H^1$ 范数下足够小的条件下，解在空间上一致地渐近收敛于平移后的粘性激波剖面。由于我们的方法直接作用于物理变量，因此不像文献 \cite{HMS03,MM99} 那样要求反导数变量属于 $L^2$ 空间。此外，对于入流情形，我们去掉了文献 \cite{HMS03} 中 $\gamma\le 3$ 的假设。我们的结果基于带平移的 $a$-收缩方法，这是该方法首次推广到边值问题。
英文摘要：
We consider the compressible barotropic Navier-Stokes equations in a half-line and study the time-asymptotic behavior toward the outgoing viscous shock wave. Precisely, we consider the two boundary problems: impermeable wall and inflow problems, where the velocity at the boundary is given as a constant state. For both problems, when the asymptotic profile determined by the prescribed constant states at the boundary and far-fields is a viscous shock, we show that the solution asymptotically converges to the shifted viscous shock profiles uniformly in space, under the condition that initial perturbation is small enough in $H^1$ norm. Since our method works on the physical variables, we do not require that the anti-derivative variables belong to $L^2$ space as in \cite{HMS03,MM99}. Moreover, for the inflow case, we remove the assumption $γ\le 3$ in \cite{HMS03}. Our results are based on the method of $a$-contraction with shifts, as the first extension of the method to the boundary value problems.
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