论文标题
关于网络上奇异扰动的对流扩散问题的运输极限
On the transport limit of singularly perturbed convection-diffusion problems on networks
论文作者
论文摘要
我们认为一维网络（度量图）上的对流扩散方程以及在消失的扩散极限中产生的传输问题。得出了内部顶点处的合适的耦合条件，以确保质量的保存以及数学能量的耗散，这使我们能够证明稳定性和稳定性。对于单个间隔和适当指定的初始条件，众所周知，对流扩散问题的解决方案将$ o（\sqrtε）$（\ l^\ infty（l^2）$ - narm-norm-norm-narm in-bixfusion $ tofusion $ε\ to 0 $）收敛于运输问题的解决方案。在本文中，我们证明了一维网络问题的相应结果。分析的主要困难是，耦合条件的数量和类型在奇异极限中发生变化，这会导致网络内部顶点的其他边界层。由于在这些网络连接处的溶液值尚不清楚A-Priori，因此渐近分析需要微妙的边界层函数选择，该函数可以处理这些内部层。
We consider singularly perturbed convection-diffusion equations on one-dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling condition at inner vertices are derived that guarantee conservation of mass as well as dissipation of a mathematical energy which allows us to prove stability and well-posedness. For single intervals and appropriately specified initial conditions, it is well-known that the solutions of the convection-diffusion problem converge to that of the transport problem with order $O(\sqrtε)$ in the $L^\infty(L^2)$-norm with diffusion $ε\to 0$. In this paper, we prove a corresponding result for problems on one-dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a-priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.
