Time-dimensional reduction and a Carleman contraction principle for an inverse initial data problem for the incompressible Navier-Stokes equations with unknown body force

1. Problem Statement

The paper addresses a challenging inverse initial data problem for the incompressible Navier–Stokes equations in a bounded domain $\Omega \subset \mathbb{R}^d$ over a time interval $(0, T)$.

• Governing Equations: The system consists of the momentum balance and incompressibility condition:
  $$\begin{cases} u_t + (u \cdot \nabla)u = -\nabla p + \mu \Delta u + f & \text{in } \Omega \times (0, T), \\ \text{div } u = 0 & \text{in } \Omega \times (0, T), \\ u = 0 & \text{on } \partial\Omega \times (0, T), \\ u(x, 0) = u_0, \quad p(x, 0) = p_0 & \text{in } \Omega. \end{cases}$$
• The Unknowns: The goal is to recover the initial velocity $u_0$ and initial pressure $p_0$.
• The Challenge: The body force $f(x)$ is unknown and time-independent. In most inverse problems, model parameters (like $f$) are assumed known; here, $f$ is an additional unknown that complicates the reconstruction.
• Available Data: The only available information is lateral boundary data measured on $\partial\Omega \times (0, T)$:
  • Normal derivative of velocity: $g = \partial_\nu u$.
  • Pressure and its normal derivative: $h = [p, \partial_\nu p]^T$.

2. Methodology

The authors propose a novel framework combining time-dimensional reduction with a Carleman-weighted contraction mapping principle. The methodology proceeds in three main stages:

A. Elimination of the Unknown Force

Since the body force $f$ is time-independent, the authors differentiate the momentum equation with respect to time ($t$). This eliminates $f$ entirely, yielding a new system for the time-derivative of the velocity ($u_t$) and pressure ($p_t$):
$$u_{tt} + (u_t \cdot \nabla)u + (u \cdot \nabla)u_t = -\nabla p_t + \mu \Delta u_t.$$
By taking the divergence of this equation and utilizing $\text{div } u = 0$, they derive a coupled system involving only $u_t$ and $p_t$, effectively decoupling the unknown force from the reconstruction process.

B. Legendre Polynomial–Exponential Time-Dimensional Reduction

The authors expand the unknown fields $u(x,t)$ and $p(x,t)$ using a specific basis:
$$u(x,t) = \sum_{n=0}^\infty u_n(x) \Psi_n(t), \quad p(x,t) = \sum_{n=0}^\infty p_n(x) \Psi_n(t),$$
where $\Psi_n(t) = e^t Q_n(t)$, and $Q_n$ are shifted Legendre polynomials on $(0, T)$.

• Truncation: The infinite series are truncated at a finite level $N$.
• Projection: The time-differentiated equations are projected onto the basis functions $\Psi_m(t)$ using a weighted inner product $\langle \cdot, \cdot \rangle_{e^{-2t}}$.
• Result: This transforms the original time-dependent inverse problem into a finite-dimensional coupled system of elliptic partial differential equations for the coefficient vectors $U_N = [u_0, \dots, u_N]^T$ and $P_N = [p_0, \dots, p_N]^T$. The boundary data is similarly projected to provide Cauchy data for these coefficients.

C. Carleman Contraction Framework

To solve the resulting nonlinear elliptic system without requiring a good initial guess (a common failure point for standard optimization methods), the authors employ a Carleman-weighted Picard iteration:

1. Regularized Operator: They define an operator $T_{\lambda, \epsilon}$ that maps a trial pair $(V, Q)$ to the minimizer of a weighted least-squares functional. The nonlinearity is "frozen" using the trial pair.
2. Weighted Norm: The minimization is performed in a space equipped with a Carleman weight $e^{2\lambda \mu_\beta(x)}$, where $\mu_\beta(x) = |x-x_0|^{-\beta}$.
3. Contraction Proof: Using Carleman estimates (inequalities that bound the solution of a PDE by its source term and boundary data), they prove that for a sufficiently large parameter $\lambda$, the operator $T_{\lambda, \epsilon}$ is a contraction on a suitable admissible set.
4. Global Convergence: By the Banach Fixed Point Theorem, the iteration converges globally to a unique fixed point, regardless of the initial guess. This fixed point approximates the solution to the reduced elliptic system.

3. Key Contributions

• Handling Unknown Body Force: The paper successfully solves an inverse problem where the body force is completely unknown, a scenario often considered intractable in standard formulations.
• Global Convergence without Initial Guess: Unlike traditional optimization approaches (e.g., Gauss-Newton) which are non-convex and sensitive to initial guesses, the proposed Carleman-Picard method guarantees global convergence to the solution of the reduced model.
• Novel Reduction Technique: The combination of time-differentiation to eliminate $f$ and the Legendre-exponential basis reduction transforms a complex parabolic inverse problem into a solvable elliptic system.
• Rigorous Analysis: The authors provide a complete theoretical proof of the contractive property of the operator and the consistency of the fixed point with the reduced system as the regularization parameter $\epsilon \to 0$.

4. Numerical Results

The authors implemented Algorithm 1 (Carleman-Picard method) and tested it on synthetic data generated from the forward Navier-Stokes equations in a 2D square domain ($\Omega = (-1, 1)^2$).

• Setup:
  • Three distinct test cases with different complex body forces and initial velocity/pressure fields.
  • Boundary data was generated and corrupted with 10% random noise.
  • Parameters: Truncation level $N=35$, Carleman weight parameters $\beta=20, \lambda=6$.
• Performance:
  • Accuracy: The method successfully reconstructed the main qualitative features (locations, signs, and shapes) of the initial velocity and pressure.
  • Error Metrics:
    • Test 1: Relative $L^2$ errors $\approx 14.5\%$ (velocity) and $23.9\%$ (pressure).
    • Test 2: Relative $L^2$ errors $\approx 17.6\%$ (velocity) and $18.7\%$ (pressure).
    • Test 3 (most complex): Errors increased to $\approx 36\%$ (velocity) and $42.8\%$ (pressure), but the method still captured the essential structures.
  • Convergence: The Picard iteration showed rapid and stable convergence, with relative residuals and iterate changes decreasing steadily. The method remained stable even with noisy data.

5. Significance and Future Directions

• Practical Relevance: This approach is highly relevant for applications (aerospace, oceanography, environmental modeling) where initial flow conditions are unknown and external forces (like wind or gravity variations) are difficult to measure or model precisely.
• Theoretical Impact: It demonstrates that Carleman estimates can be effectively adapted to nonlinear fluid dynamics problems, overcoming structural assumptions that limited previous applications to simpler parabolic equations.
• Future Work: The authors suggest extending the method to partial boundary data, refining convergence rates for fully discrete schemes, and applying the framework to 3D geometries and more general fluid systems.

In summary, this paper presents a robust, globally convergent numerical method for a difficult inverse problem in fluid dynamics, successfully bypassing the need for prior knowledge of the body force and avoiding the pitfalls of local minima common in nonlinear inverse problems.