Observers for compressible Navier-Stokes equation

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