We consider the inverse initial data problem for the compressible anisotropic Navier-Stokes equations, where the goal is to reconstruct the initial velocity field from lateral boundary observations. This problem arises in applications where direct measurements of internal fluid states are unavailable. We introduce a novel computational framework based on Legendre time reduction, which projects the velocity field onto an exponentially weighted Legendre basis in time. This transformation reduces the original time-dependent inverse problem to a coupled, time-independent elliptic system. The resulting reduced model is solved iteratively using a Picard iteration and a stabilized least-squares formulation under noisy boundary data. Numerical experiments in two dimensions confirm that the method accurately and robustly reconstructs initial velocity fields, even in the presence of significant measurement noise and complex anisotropic structures. This approach offers a flexible and computationally tractable alternative for inverse modeling in fluid dynamics with anisotropic media.

翻译：本文研究可压缩各向异性Navier-Stokes方程的反初值问题，其目标是通过侧向边界观测数据重构初始速度场。该问题产生于无法直接测量流体内部状态的实际应用中。我们提出了一种基于Legendre时间降维的新型计算框架，将速度场投影到指数加权的Legendre时间基上。该变换将原始的含时反问题转化为耦合的、与时间无关的椭圆型方程组。所得降维模型在含噪声边界数据下，通过Picard迭代与稳定化最小二乘公式进行迭代求解。二维数值实验证实，即使在显著测量噪声和复杂各向异性结构存在的情况下，该方法仍能准确且鲁棒地重构初始速度场。该研究为各向异性介质流体动力学中的反问题建模提供了一种灵活且计算可行的新途径。