基于复几何光学解的两类非稳态扩散方程反问题研究

The idea of the complex geometrical optics solutions, which is raised by the famous mathematicians, Professor Uhlmann and Professor Sylvester, is applied to study the uniqueness and stability by the construction of a special solution of the Schrodinger equation as well as problem-specific nonlinear Fourier analyses. Although the method plays an important role in the study of the inverse problems of hyperbolic and elliptic partial differential equations and has obtained many significantly innovative results, the use of this method in the study of parabolic equations is just at the beginning. .The coefficient inverse problems and inverse boundary problems of nonsteady- state diffusion equations are major difficult inverse problems for traditional methods. The project subjects to investigate these problems, including coefficient inverse problems of classical diffusion equations and time-fractional diffusion equations, inverse boundary problems for classical diffusion equations, based on the idea of the CGO solutions. In order to obtain the uniqueness and conditional stability which depended upon the measured boundary data and different types of boundary conditions, we construct CGO solutions to the specific problems. Furthermore, provide regularization methods with the same convergence rates to the stability results by applying the proposed CGO solutions and the theory of pseudo- differential operator, with an emphasis on modifying the symbols, search for the corresponding regularized solutions and simulate the approximation numerically.

复几何光学解是近年来由Uhlmann教授等著名数学家提出的研究反问题的一种新的重要思想，通过构造Schrodinger方程的特殊解构造非线性Fourier变换，从而研究唯一性和条件稳定性。这种方法在研究双曲方程和椭圆方程反问题中发挥了巨大作用，得到了许多重要的创新性结果，但该方法对抛物型偏微分方程反问题的研究才刚刚起步。. 非稳态扩散方程的系数反问题和边界辨识问题，是反问题研究的难点，用已有的传统工具很难突破。本课题拟用复几何光学解这一新的思想和工具对这两类困难问题开展研究。构造出满足具体问题的复几何光学解，针对不同类型的边界测量条件，研究问题的唯一性和条件稳定性；进而通过已得到的复几何光学解，结合拟微分算子理论，重点是修改振幅函数，构造出与条件稳定性结果具相同收敛阶数的正则化方法，并在此基础上对这两类反问题进行数值模拟。

本项目主要研究内容包括以下三部分： （1）耦合Schrödinger方程组和波方程组的系数识别问题。通过观测解的一个分量在方程所在区域的非空开子集中的性质，获得对数型稳定性结果。在研究过程中采用了Calerman估计与Fourier-Bros-Iagolnitzer变换结合的方法。（2）核基逼近方法求解时空分数阶扩散方程反问题。利用Fourier变换技术和分数阶扩散方程基本解的性质数值模拟方程的基本解，并采用合理的方式选取配置点和源点，以减少循环，加快运算速度，并保证方法的有效性的精确性。同时研究核基逼近方法的收敛性估计。（3）二维及三维区域的趋化Navier-Stokes系统解的整体存在性、长时间行为，唯一性及稳定性。其中，（2）、（3）两部分的成果已发表在Appl. Math. Comput., Comput. Math. Appl., Math. Models Methods Appl. Sci., J. Differential Equations等期刊上，（1）部分的研究结果已投稿。