  This paper concerns the stabilizing effect of viscosity on the vortex sheets.
It is found that although a vortex sheet is not a time-asymptotic attractor for
the compressible Navier-Stokes equations, a viscous wave that approximates the
vortex sheet on any finite time interval can be constructed explicitly, which
is shown to be time-asymptotically stable in the $ L^\infty $-space with small
perturbations, regardless of the amplitude of the vortex sheet. The result
shows that the viscosity has a strong stabilizing effect on the vortex sheets,
which are generally unstable for the ideal compressible Euler equations even
for short time [26,8,1]. The proof is based on the $ L^2 $-energy method.In
particular, the asymptotic stability of the vortex sheet under small spatially
periodic perturbations is proved by studying the dynamics of these spatial
oscillations. The first key point in our analysis is to construct an ansatz to
cancel these oscillations. Then using the Galilean transformation, we are able
to find a shift function of the vortex sheet such that an anti-derivative
technique works, which plays an important role in the energy estimates.
Moreover, by introducing a new variable and using the intrinsic properties of
the vortex sheet, we can achieve the optimal decay rates to the viscous wave.
