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Continuous Data Assimilation for the Navier-Stokes Equations with Nonlinear Slip Boundary Conditions

W. C. Wu, H. Y. Dong, K. Wang

TL;DR

This work addresses data assimilation for the two-dimensional Navier-Stokes equations with nonlinear slip boundary conditions under incomplete information, including missing initial data and unknown viscosity. The authors develop a regularized variational CDA framework, prove well-posedness and exponential convergence for missing initial data, and extend the analysis to viscosity recovery, complemented by finite element error estimates. A novel integration of partial evolutionary tensor neural networks (pETNNs) is proposed to replace observations with predictive data, enabling similar accuracy at reduced computational cost. Numerical experiments on a unit square, a circular cylinder, and a bifurcated blood-flow model validate the theoretical results and illustrate the efficiency gains from the pETNN-enhanced CDA approach.

Abstract

This paper focuses on continuous data assimilation (CDA) for the Navier-Stokes equations with nonlinear slip boundary conditions. CDA methods are typically employed to recover the original system when initial data or viscosity coefficients are unknown, by incorporating a feedback control term generated by observational data over a time period. In this study, based on a regularized form derived from the variational inequalities of the Navier-Stokes equations with nonlinear slip boundary conditions, we first investigate the classical CDA problem when initial data is absent. After establishing the existence, uniqueness and regularity of the solution, we prove its exponential convergence with respect to the time. Additionally, we extend the CDA to address the problem of missing viscosity coefficients and analyze its convergence order, too. Furthermore, utilizing the predictive capabilities of partial evolutionary tensor neural networks (pETNNs) for time-dependent problems, we propose a novel CDA by replacing observational data with predictions got by pETNNs. Compared with the classical CDA, the new one can achieve similar approximation accuracy but need much less computational cost. Some numerical experiments are presented, which not only validate the theoretical results, but also demonstrate the efficiency of the CDA.

Continuous Data Assimilation for the Navier-Stokes Equations with Nonlinear Slip Boundary Conditions

TL;DR

This work addresses data assimilation for the two-dimensional Navier-Stokes equations with nonlinear slip boundary conditions under incomplete information, including missing initial data and unknown viscosity. The authors develop a regularized variational CDA framework, prove well-posedness and exponential convergence for missing initial data, and extend the analysis to viscosity recovery, complemented by finite element error estimates. A novel integration of partial evolutionary tensor neural networks (pETNNs) is proposed to replace observations with predictive data, enabling similar accuracy at reduced computational cost. Numerical experiments on a unit square, a circular cylinder, and a bifurcated blood-flow model validate the theoretical results and illustrate the efficiency gains from the pETNN-enhanced CDA approach.

Abstract

This paper focuses on continuous data assimilation (CDA) for the Navier-Stokes equations with nonlinear slip boundary conditions. CDA methods are typically employed to recover the original system when initial data or viscosity coefficients are unknown, by incorporating a feedback control term generated by observational data over a time period. In this study, based on a regularized form derived from the variational inequalities of the Navier-Stokes equations with nonlinear slip boundary conditions, we first investigate the classical CDA problem when initial data is absent. After establishing the existence, uniqueness and regularity of the solution, we prove its exponential convergence with respect to the time. Additionally, we extend the CDA to address the problem of missing viscosity coefficients and analyze its convergence order, too. Furthermore, utilizing the predictive capabilities of partial evolutionary tensor neural networks (pETNNs) for time-dependent problems, we propose a novel CDA by replacing observational data with predictions got by pETNNs. Compared with the classical CDA, the new one can achieve similar approximation accuracy but need much less computational cost. Some numerical experiments are presented, which not only validate the theoretical results, but also demonstrate the efficiency of the CDA.

Paper Structure

This paper contains 17 sections, 9 theorems, 120 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equations and functional spaces
  4. Regularization approximation
  5. Classical Continuous Data Assimilation
  6. Problem with initial data missing
  7. Problem with viscosity coefficient missing
  8. Finite Element Approximation
  9. Numerical Experiment
  10. Flow in simple rectangular domain
  11. Flow around a circular cylinder
  12. Bifurcated Blood Flow Model
  13. Continuous Data Assimilation based on pETNNs
  14. Partial Evolutionary Tensor Neural Networks
  15. Continuous Data Assimilation based on pETNNs
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $u_0 \in H, f \in L^2([0,T];V^*)$ and $g \in L^2([0,T];L^2(S))$ with $g \geq 0$. Then the problem (23) admits a unique solution such that Furthermore, if $u_0\in \mathcal{D}(A),\ f\in L^\infty([0,T];H),\ \int_{S}g\sqrt{u_0^2+\varepsilon^2}\leq c$ and $g\in H^\frac{1}{2}(S)$ with $g\geq 0$, then, for a fixed $T^*>0$, there holds where $t+T^*\leq T$ and $G$ is defined as

Figures (7)

  • Figure 5: Domain and boundary conditions.
  • Figure : FEM CDA
  • Figure : FEM CDA
  • Figure : FEM CDA
  • Figure : FEM CDA
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 5 more