一类抛物型界面问题的正则性分析

Regular Analysis of a Class of Parabolic Interface Problems

摘要: 界面问题用于各种工程应用和物理、化学、生物现象的建模，特别是涉及具有不同扩散性、密度、渗透性或导电性的多种不同材料的现象，其在界面上由一定条件耦合。本文考虑具有溶解质输运的线性两相流模型，分别在相界面处耦合非完美界面条件和Henry界面条件。由于解在界面上的跳跃使得解在各自材料区域上具有比在整个区域上更高的正则性，针对这类界面问题的正则性分析，本文给出一个完整的泛函分析过程，采用De Giorgi迭代方法证明该模型弱解的相关性质，进而证明弱解及其梯度的Hölder连续性。此外，对于Henry界面问题，本文给出了梯度的L^q估计(存在q > 2)。

Abstract: Interface problems are used for various engineering applications and modeling of physical, chemi-cal, and biological phenomena, especially those involving a number of different materials with dif-ferent diffusion, density, permeability or conductivity, which are coupled by certain conditions at the interface. In this paper, a linear two-phase flow model with solute transport is considered, cou-pling imperfect interface condition and Henry interface condition at the phase interface, respec-tively. Since the jump of the solution at the interface makes the solution have higher regularity in the respective material region than on the whole region, for the regularity analysis of such interface problems, this paper presents a complete functional analysis process, and uses the De Giorgi itera-tion method to prove the correlation properties of the weak solution of the model, and then prove the Hölder continuity of the weak solution and its gradient. In addition, for the Henry interface problem, this paper gives the L^q estimation of the gradient (q > 2 is present).