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

• DOI: 10.48550/arxiv.2310.12747
• Publication Date: 2023-01-01
• Authors: Lingjun Liu; Shu Wang; Lingda Xu
• Keywords: Zero (linguistics); Time derivative

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