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

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 $T$.