Asymptotic stability of planar entropy wave for 3-d Navier-Stokes equations in Eulerian coordinates

Abstract

We investigate the large-time asymptotic behavior toward the planar entropy wave for the three-dimensional Navier-Stokes equations in Eulerian coordinates, considering two types of initial perturbations -- with and without the assumption that the integral of the initial perturbation is zero. Generic perturbations generate diffusion waves, and structural conditions fail for multi-dimensional Navier-Stokes equations in Eulerian coordinates. These two aspects have posed significant challenges and left the problem unresolved for years. On one hand, since \cite{LX}, the study of the entropy wave has been based on the left-right structural conditions. Without these structural conditions, the decay rates of lower-order terms become too slow to close the {\it a priori} assumption. On the other hand, the presence of diffusion waves yields problematic error terms in the perturbation system. In this work, we introduce a new transformation to ensure that both left-right structural conditions hold for the perturbation system. Additionally, using the fact that the derivative of the entropy wave maintains a fixed sign, we employ well-designed weighted energy estimates to control the slowly decaying terms. This enables us to establish asymptotic stability and derive the optimal decay rate. Furthermore, we address the case of initial perturbations with the zero mass condition and obtain the optimal decay rate by additionally developing a Poincar\'e type inequality and a key cancellation.