Optimal decay rates to the contact wave for 1-D compressible Navier-Stokes equations

This paper investigates the decay rates of the contact wave in one-dimensional Navier-Stokes equations. We study two cases of perturbations, with and without zero mass condition, i.e., the integration of initial perturbations is zero and non-zero, respectively. For the case without zero mass condition, we obtain the optimal decay rate $(1+t)^{-\frac{1}{2}}$ for the perturbation in $L^\infty$ norm, which provides a positive answer to the conjecture in \cite{HMX}. We applied the anti-derivative method, introducing the diffusion wave to carry the initial excess mass, diagonalizing the integrated system, and estimating the energy of perturbation in the diagonalized system. Precisely, due to the presence of diffusion waves, the decay rates for errors of perturbed system are too poor to get the optimal decay rate. We find the dissipation structural in the diagonalized system, see \cref{ds}. This observation makes us able to fully utilize the fact that the sign of the derivative of the contact wave is invariant and to control the terms with poor decay rates in energy estimates. For the case with zero mass condition, there are also terms with poor decay rates. In this case, note that there is a cancellation in the linearly degenerate field so that the terms with poor decay rates will not appear in the second equation of the diagonalized system. Thanks to this cancellation and a Poincar\'e type of estimate obtained by a critical inequality introduced by \cite{HLM}, we get the decay rate of $\ln^{\frac{1}{2}} (2+t)$ for $L^2$ norm of anti-derivatives of perturbation and $(1+t)^{-\frac{1}{2}}\ln^{\frac{1}{2}}(2+t)$ for the $L^2$ norm of perturbation itself, the decay rates are optimal, which is consistent with the results obtained by using pointwise estimate in \cite{XZ} for the system with artificial viscosity.