Local exact Lagrangian controllability for 1D barotropic compressible Navier--Stokes equations

TL;DR

This paper demonstrates the exact controllability of a 1D viscous compressible flow by external forces, allowing precise movement of fluid particles between specified subintervals within a finite time.

Contribution

It introduces a method to achieve local exact Lagrangian controllability for the 1D barotropic Navier--Stokes equations using localized external forces.

Findings

Constructed external force achieves particle movement between subintervals

Established controllability within finite time

Applicable to flows with homogeneous boundary conditions

Abstract

We consider a viscous compressible barotropic flow in the interval $[0,\pi]$ with homogeneous Dirichlet boundary conditions for the flow velocity and a constant rest state as initial data. Given two sufficiently close subintervals $I=[\alpha_1,\alpha_2]$ and $J=[\beta_1,\beta_2]$ of $(0,1)$, a nonempty open set $\omega \subset (1,\pi)$, and $T>0$, we construct an external force $f$ supported in $\omega$ acting on the momentum equation such that the corresponding flow map moves the fluid particles initially occupying $I$ exactly onto $J$ in time $T$.