论文标题
通过数值近似构建弱解决方案以压缩Navier-稳定的方程式具有一般流入/流出边界条件
Construction of weak solutions to compressible Navier--Stokes equations with general inflow/outflow boundary conditions via a numerical approximation
论文作者
论文摘要
通过数值方法（包括对收敛的严格证明）为可压缩的Navier-Stokes方程的弱解决方案的构建是短的，到目前为止，仅适用于karper中建议的一种唯一的数值方案[{\ em numer。数学。}，125（3）：441--510，2013]对于无滑移边界条件和具有绝热系数$γ> 3 $的等性压力。在这里，我们考虑了一般非零流出流量边界条件的相同问题，从应用程序的角度来看，这绝对是更合适的设置，但是就弱解决方案的存在而言，这基本上更为参与。在这种情况下，最近有一些弱解决方案存在的证据，但是没有一个是通过数值方法执行的。本文的目的是填补这一空白。 在连续水平上存在弱解的存在需要几种功能和谐波分析的工具以及差异几何形状，它们的数值对应物尚不清楚。因此，我们的主要策略在于在其变异形式的剩余形式中重写数值方案，并应用和/或适应理论分析中开发的工具的新变异表述。除了新的结果外，数值和理论分析之间的协同作用是本文的主要独创性。
The construction of weak solutions to compressible Navier-Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper [{\em Numer. Math.}, 125(3) : 441--510, 2013] for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient $γ>3$. Here we consider the same problem for the general non zero inflow-outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper.
