Observers for compressible Navier-Stokes equation

TL;DR

This paper develops an observer for a multi-dimensional compressible Navier-Stokes fluid model, enabling estimation of density and velocity from velocity observations, with proven convergence for linearized systems and promising numerical results for nonlinear cases.

Contribution

It introduces a novel observer exploiting symmetries to estimate unobserved variables from partial velocity data in compressible fluid models.

Findings

Observer converges to true state at any desired rate for linearized, fully observed systems.

Numerical results show convergence in nonlinear systems and partial domain observations.

Method effectively estimates density and velocity from limited velocity observations.

Abstract

We consider a multi-dimensional model of a compressible fluid in a bounded domain. We want to estimate the density and velocity of the fluid, based on the observations for only velocity. We build an observer exploiting the symmetries of the fluid dynamics laws. Our main result is that for the linearised system with full observations of the velocity field, we can find an observer which converges to the true state of the system at any desired convergence rate for finitely many but arbitrarily large number of Fourier modes. Our one-dimensional numerical results corroborate the results for the linearised, fully observed system, and also show similar convergence for the full nonlinear system and also for the case when the velocity field is observed only over a subdomain.