Long-time behavior toward composite wave of shocks for 3D barotropic navier-stokes system

We consider the barotropic Navier-Stokes system in three space dimensions with periodic boundary condition in the transversal direction. We show the long-time behavior of the 3D barotropic Navier-Stokes flow perturbed from a composition of two shock waves with suitably small amplitudes. We prove that the perturbed Navier-Stokes flow converges, uniformly in space, towards a composition of two planar viscous shock waves as time goes to infinity, up to dynamical shifts. This is the first result on time-asymptotic stability of composite wave of two shocks for multi-D Navier-Stokes system. The main part of proof is based on the method of a-contraction with shifts.