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Data assimilation for the barotropic Navier-Stokes system

Eduard Feireisl

Abstract

We consider a continuous data assimilation method for the barotropic Navier--Stokes system. The observed solution is supposed to be bounded on the whole time period of observation, while the synchronized solution, usually provided by a numerical method, belongs to the class of dissipative solutions that is considerably larger than the class of conventional weak solutions. A complete synchronization is shown on any compact prediction interval provided the nudging parameters are chosen appropriately.

Data assimilation for the barotropic Navier-Stokes system

Abstract

We consider a continuous data assimilation method for the barotropic Navier--Stokes system. The observed solution is supposed to be bounded on the whole time period of observation, while the synchronized solution, usually provided by a numerical method, belongs to the class of dissipative solutions that is considerably larger than the class of conventional weak solutions. A complete synchronization is shown on any compact prediction interval provided the nudging parameters are chosen appropriately.
Paper Structure (12 sections, 2 theorems, 74 equations)

This paper contains 12 sections, 2 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and the main result
  3. Data sampling -- interpolation operators
  4. Synchronized system - data assimilation
  5. Main result
  6. Preliminary results concerning the observed and synchronized system
  7. Conditional regularity for the observed system
  8. Relative energy inequality for the synchronized system
  9. Bounds on the relative energy in the data assimilation period
  10. Uniform bounds depending on the data
  11. Bounds on the interpolation error
  12. Control in the prediction (forecast) period

Key Result

Theorem 2.2

Let $\Omega \subset R^3$ be a bounded domain of class $C^4$. Let the pressure--density equation of state $p=p(\varrho)$ satisfy the hypotheses S4a, S4A, together with the growth restriction SS3a, where $\gamma > \frac{6}{5}$. Suppose the observed solution $(r, {\bf U})$ belongs to the regularity cla Let $(\varrho, {\bf u})$ be a dissipative solution of the synchronized system S1--S3 in the sense o

Theorems & Definitions (6)

  • Definition 2.1: Dissipative solution
  • Theorem 2.2: Synchronization for the compressible Navier-Stokes system
  • Remark 2.3
  • Proposition 3.1: Conditional regularity
  • Remark 4.1
  • Remark 5.1