Optimal decay rate to the contact discontinuity for Navier–Stokes equations under generic perturbations

This paper investigates the large-time asymptotic behavior of contact waves in 1-D compressible Navier–Stokes equations. We derive the optimal decay rate for generic initial perturbations, meaning the perturbation’s integral does not need to be zero. It is well-known that generic perturbations in Navier–Stokes equations generate diffusion waves, implying that the optimal decay rate for contact waves in the $L^{\infty }$-norm is ${(1+t)}^{-1/2}$. However, the presence of diffusion waves introduces error terms, leading to energy growth in the anti-derivatives of the perturbations. Furthermore, studying contact waves depends on certain structural conditions, which hold for the original system but not for its derivative systems. This makes it challenging to obtain accurate estimates for the energy of the derivatives.

In this paper, we refine the estimates for both anti-derivatives and the original perturbations. We then introduce an innovative transformation to ensure that the structural conditions continue to hold for the system of derivatives. With this approach, we achieve better estimates for the derivatives, leading to the optimal decay rates. This result improves upon the well-known findings of Huang et al. (2008), and the method has the potential for application in more general systems.