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Continuous data assimilation applied to the Rayleigh-Benard problem for compressible fluid flows

Eduard Feireisl, Wladimir Neves

TL;DR

This work proves that a continuous data assimilation (nudging) scheme applied to the 3D compressible Navier–Stokes–Fourier system in a rotating Rayleigh–Bénard setting yields a rigorous tracking result: for sufficiently small Mach number and large rotation/stratification, the synchronized solution converges to the observed solution on a finite horizon in the weak-solution framework. The analysis hinges on (i) identifying the asymptotic target system as a 2D incompressible Boussinesq-type model, (ii) establishing global well-posedness and tracking for the target and synchronized-target problems, and (iii) a relative-energy (Bregman) method to compare the synchronized NSF system with the target and observed states. By combining the asymptotic limit with a Grönwall argument, the authors demonstrate that the differences between observed and synchronized variables can be arbitrarily small under suitable choices of the nudging parameters $\Lambda$ and $\delta$ and small data perturbations. This provides a first rigorous confirmation that continuous data assimilation can stabilize and predict the evolution of physically realistic compressible flows under Rayleigh–Bénard forcing in a rotating frame, within a weak-solution setting. The results have potential implications for mathematically grounded data assimilation in geophysical and astrophysical convection problems where compressibility, rotation, and heat conduction are essential.

Abstract

We apply a continuous data assimilation method to the Navier-Stokes-Fourier system governing the evolution of a compressible, rotating and thermally driven fluid. A rigorous proof of the tracking property is given in the asymptotic regime of low Mach and high Rossby and Froude numbers. Large data in the framework of weak solutions are considered.

Continuous data assimilation applied to the Rayleigh-Benard problem for compressible fluid flows

TL;DR

This work proves that a continuous data assimilation (nudging) scheme applied to the 3D compressible Navier–Stokes–Fourier system in a rotating Rayleigh–Bénard setting yields a rigorous tracking result: for sufficiently small Mach number and large rotation/stratification, the synchronized solution converges to the observed solution on a finite horizon in the weak-solution framework. The analysis hinges on (i) identifying the asymptotic target system as a 2D incompressible Boussinesq-type model, (ii) establishing global well-posedness and tracking for the target and synchronized-target problems, and (iii) a relative-energy (Bregman) method to compare the synchronized NSF system with the target and observed states. By combining the asymptotic limit with a Grönwall argument, the authors demonstrate that the differences between observed and synchronized variables can be arbitrarily small under suitable choices of the nudging parameters and and small data perturbations. This provides a first rigorous confirmation that continuous data assimilation can stabilize and predict the evolution of physically realistic compressible flows under Rayleigh–Bénard forcing in a rotating frame, within a weak-solution setting. The results have potential implications for mathematically grounded data assimilation in geophysical and astrophysical convection problems where compressibility, rotation, and heat conduction are essential.

Abstract

We apply a continuous data assimilation method to the Navier-Stokes-Fourier system governing the evolution of a compressible, rotating and thermally driven fluid. A rigorous proof of the tracking property is given in the asymptotic regime of low Mach and high Rossby and Froude numbers. Large data in the framework of weak solutions are considered.
Paper Structure (17 sections, 2 theorems, 97 equations)

This paper contains 17 sections, 2 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Continuous data assimilation problem
  4. Preliminaries, main results
  5. Hypotheses imposed on constitutive relations
  6. Weak solutions to the observed system
  7. Weak solutions to the (forced) synchronized system
  8. Main results
  9. Asymptotic limit of the observed NSF system as $\varepsilon \to 0$
  10. A data assimilation method for the target system
  11. Global existence for the target problem
  12. Tracking property for the target system
  13. Asymptotic limit of the synchronized NSF system as $\varepsilon \to 0$
  14. Ansatz in the relative energy inequality
  15. Additional terms in the relative energy inequality
  16. ...and 2 more sections

Key Result

Theorem 2.4

Suppose the thermodynamic functions $p$, $e$, and $s$ satisfy the hypotheses w9--w14a, and the transport coefficients $\mu$, $\eta$, and $\kappa$ are continuously differentiable functions of the temperature satisfying w16. Let $(\varrho, \vartheta, {\bf u})$ be a weak solution of the observed NSF sy and such that whenever

Theorems & Definitions (7)

  • Remark 1.1
  • Remark 2.1
  • Definition 2.2: Weak solution to the NSF system
  • Remark 2.3
  • Theorem 2.4: Tracking property
  • Remark 3.1
  • Proposition 4.1