发表机构
Huazhong University of Science and Technology; Hubei Key Laboratory of Engineering Modeling and Scientific Computing（华中科技大学; 湖北省工程建模与科学计算重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文在共形横截各向异性流形上，针对抛物型方程中时间相关势的反演问题，建立了双对数稳定性估计，通过部分输入-输出映射和均匀稳定性假设，结合高斯束拟模与Carleman估计等方法，证明了恢复的稳定性。
AI 中文摘要
我们在共形横截各向异性流形上建立了抛物型方程$(c^{-1}\partial_t-\Delta_g+q)u=0$中时间相关势恢复的双对数稳定性估计。部分输入-输出映射将初始状态和支撑在指定开集中的横向Dirichlet数据与另一开集上的终端状态和Neumann迹相关联。这些集合分别包含由极限Carleman权重确定的正、负边界部分。测量差异是两个此类映射之差的算子范数，其值域具有改进的Sobolev正则性。我们在非切向测地线族上对衰减测地线射线变换的均匀稳定性假设下证明了稳定性。当横向流形为简单流形以及在某些非简单流形上的正则族时，该假设得到验证。证明使用了具有均匀$O(h^{1/2})$集中估计的高斯束拟模、由边界Carleman估计得到的几何光学解，以及小衰减下测地线射线变换的稳定性。Hilbert空间值Fourier变换的定量解析延拓和Sobolev插值产生了双对数模量。
英文摘要
We establish a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation $(c^{-1}\partial_t-Δ_g+q)u=0$ on a conformally transversally anisotropic manifold. The partial input--output map associates an initial state and lateral Dirichlet data supported in a prescribed open set with the terminal state and the Neumann trace on another open set. These sets contain the positive and negative boundary parts determined by a limiting Carleman weight, respectively. The measurement discrepancy is the operator norm of the difference of two such maps, whose range has improved Sobolev regularity. We prove stability under a uniform stability assumption for the attenuated geodesic ray transform on a family of non-tangential geodesics. This assumption is verified when the transversal manifold is simple and for regular families on certain non-simple manifolds. The proof uses Gaussian beam quasimodes with a uniform $O(h^{1/2})$ concentration estimate, geometric optics solutions obtained from boundary Carleman estimates, and stability of the geodesic ray transform for small attenuation. Quantitative analytic continuation for Hilbert-space-valued Fourier transforms and Sobolev interpolation yield the double-logarithmic modulus.
Comments35 pages, no figures
