AI 中文总结
该研究针对共形横截各向异性流形上的对流扩散方程反问题，利用部分输入-输出算子，在1-形式与函数的衰减测地射线变换单射的条件下，唯一确定时变对流项和密度系数。
AI 中文摘要
我们研究在维数至少为3的某类紧致黎曼流形上，从部分数据恢复对流扩散方程的时变对流项和密度系数的反问题。我们证明，部分输入-输出算子的知识可唯一确定这两个系数，且其确定程度在自然规范变换下等价。我们的几何背景是共形横截各向异性流形，即带边界的紧致黎曼流形，可共形嵌入到欧氏直线与横截流形的乘积中。此外，我们假设横截流形上的1-形式和函数的衰减测地射线变换均为单射。
英文摘要
We study an inverse problem of recovering a time-dependent convection term and density coefficient of the convection-diffusion equation from partial data on a certain type of compact Riemannian manifold of dimension at least three. We prove that the knowledge of a partial input-output operator determines both coefficients uniquely up to a natural gauge. Our geometric setting is conformally transversally anisotropic manifolds, that is, compact Riemannian manifolds with boundary that are conformally embedded into a product of the Euclidean line and a transversal manifold. Additionally, we assume that the attenuated geodesic ray transforms of one-forms and functions are both injective on the transversal manifold.
