The inverse initial data problem for anisotropic Navier-Stokes equations via Legendre time reduction method

TL;DR

This work tackles the inverse initial data problem for compressible anisotropic Navier–Stokes equations by introducing a Legendre time reduction that projects the time dependence onto an exponentially weighted Legendre basis, transforming the original time-dependent problem into a sequence of time-independent elliptic systems. A Picard iteration, coupled with a stabilized quasi-reversibility least-squares approach, solves these reduced models under noisy boundary data, enabling robust reconstruction of the initial velocity field from lateral boundary measurements. Numerical experiments in 2D demonstrate accurate recovery of complex, high-contrast, anisotropic initial states even with 10% boundary noise, highlighting the method's practical potential for anisotropic inverse fluid problems. The framework provides a flexible, computationally tractable pathway toward stable inverse modeling in anisotropic media, with future work aimed at theoretical convergence guarantees via Carleman techniques and extensions to broader settings.

Abstract

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.

Figures (4)

• Figure 1: Test 1: The true and computed solutions to the inverse initial data problem. The first row shows the true velocity fields and the logarithm of the error between consecutive iterations. The second row displays the computed velocity fields corresponding to the reconstruction.
• Figure 2: Test 2: The true and computed solutions to the inverse initial data problem. The first row shows the true velocity fields and the logarithm of the error between consecutive iterations. The second row displays the computed velocity fields corresponding to the reconstruction.
• Figure 3: Test 3: The true and computed solutions to the inverse initial data problem. The first row shows the true velocity fields and the logarithm of the error between consecutive iterations. The second row displays the computed velocity fields corresponding to the reconstruction.
• Figure 4: Test 1: effect of the time basis in the time-dimensional reduction. (a,d) Ground-truth initial fields $u_0^{\mathrm{true},1}$ and $u_0^{\mathrm{true},2}$. (b,e) Reconstructions using the unweighted Legendre basis $\{Q_n\}_{n\ge0}$ in $L^2(0,T)$. (c,f) Reconstructions using the exponentially weighted basis $\{\Psi_n\}_{n\ge0}$ with $\Psi_n=e^{t}Q_n$ in $L^2_{e^{-2t}}(0,T)$. Under identical $N$ and regularization, the weighted basis yields sharper localization and amplitudes with fewer artifacts than the unweighted alternative.

Theorems & Definitions (11)

• Proposition 1
• Remark 1: The importance of the weight $e^t$
• Remark 2
• Remark 3
• Theorem 1: Asymptotic commutation theorem
• proof
• Remark 4: Regularity in Theorem \ref{['thm1']} and observed performance
• Remark 5: Asymptotic commutation
• Remark 6
• Remark 7