Global Recovery from Local Data: Interior Nudging for 2D Navier-Stokes equations in a Physical Domain

TL;DR

This work shows that full recovery of the 2D Navier–Stokes flow on a bounded domain can be achieved using only interior observations, without boundary data, by a continuous data assimilation approach with interpolant nudging. A rigorous synchronization theorem provides explicit conditions on the nudging parameter μ, coarse observation density H, and the geometric distance δ to guarantee exponential convergence of the assimilated state to the true solution, with a δ-constraint that scales like δ ≲ ν^{1/2}. Computational experiments across complex geometries demonstrate robust global synchronization from interior data, while boundary measurements are largely uninformative, reinforcing the practicality of interior sensing. The results have implications for sensor placement in real-world DA applications and motivate extensions to 3D flows and adaptive observation strategies, informed by boundary-layer physics.

Abstract

In many real-world applications of data assimilation (DA), the strategic placement of observers is crucial for effective and efficient forecasting. Motivated by practical constraints in sensor deployment, we show that global recovery of the flow field can be achieved using observations available only in a subregion of the domain, possibly far from the boundary. We focus on the two-dimensional incompressible Navier-Stokes equations posed in a bounded physical domain with Dirichlet boundary conditions. Building on the continuous data assimilation framework of Azouani, Olson, and Titi (2014), we rigorously prove that the assimilated solution converges globally to the true solution under suitable conditions on the nudging parameter, spatial resolution, and the geometry of the observation region, specifically, when the maximum distance from any point in the domain to the observational subregion is bounded by a constant multiple of \( ν^{1/2} \) (in terms of scaling). Our computational results, conducted via finite element methods over complex geometries, support the theoretical findings and reveal even greater robustness in practice. Specifically, synchronization with the true solution is achieved even when the observational subregion lies farther from the rest of the domain than the theoretical threshold permits. Across all three tested scenarios, the local nudging algorithm performs comparably to full-domain assimilation, reaching global accuracy up to machine precision. Interestingly, observational data near the boundary are found to be largely uninformative. This demonstrates that full observability is not necessary: carefully chosen interior observations, even far from the boundary, can suffice.

Figures (9)

• Figure 1: Domain mesh setup: from local coarse observations to global DNS mesh.
• Figure 2: Local coarse mesh construction within the domain $\Omega$, showing three sub-domains. Region $1$$(\Omega_1)$: a box-shaped region centered around the flat plate, Region $2$$(\Omega_2)$: the interior region between the top and bottom wall layers and the box-shaped region. And Region $3$$(\Omega_3)$: the boundary layers near the top and bottom walls.
• Figure 3: Comparison of results using the local data assimilation algorithm applied to the flow over a flat obstacle with data collected from different regions of the domain: $\Omega_1$, $\Omega_2$, and $\Omega_3$.
• Figure 4: Relative error when applying local DA to $\Omega_2$ with different $l$.
• Figure 5: Relative error when applying DA to $\Omega$ with different $\mu$ values.

Theorems & Definitions (7)

• Theorem 2.1: Existence and Uniqueness of Strong Solutions in $2D$
• Theorem 2.2
• Proposition 2.3: Uniform Gronwall Inequality
• Theorem 4.1
• Remark 4.2
• proof
• Remark 4.3