Asymptotic stability of steady states for the compressible Navier-Stokes-Riesz system in the presence of vacuum

Abstract: We consider a one-dimensional physical vacuum free boundary problem on the compressible Navier-Stokes-Riesz system for an attractive Riesz potential $|x|^{2s-1}/(2s-1)$ with $0<s<1/2$. It is proved that for the adiabatic constant $γ$ satisfying $2(1-s)<γ<1+2s/3$ under the additional condition that $3/8<s<1/2$, there exists a unique global-in-time strong solution. Specifically, we establish the Lyapunov-type stability of the compactly supported steady states in the Lagrangian coordinates and we also obtain the time rate of convergence for the strong solution to steady states with the same mass in weighted Sobolev spaces where the weights indicate the behavior of solutions near the vacuum free boundary. The difficulties and challenges in the proof are caused not only by the degeneracy due to the vacuum free boundary but also by the non-local feature of the Riesz potential.