Asymptotic stability of planar entropy wave for 3-d Navier-Stokes equations in Eulerian coordinates
Ren-Jun Duan, Feimin Huang, Rui Li, Lingda Xu
TL;DR
The paper proves nonlinear asymptotic stability of planar entropy waves for the 3-D Navier–Stokes equations in Eulerian coordinates under small perturbations, addressing both nonzero and zero mass initial data. It introduces a novel ansatz and a decomposition into zero and nonzero Fourier modes, along with a transformation that enforces left–right structural conditions for the perturbation system and a weighted energy framework to control slowly decaying terms. For nonzero mass, the perturbation decays at the optimal rate $(1+t)^{-1/2}$ with exponential decay of nonzero modes; for zero mass, the zero-mode decay improves to $(1+t)^{-3/4}$ up to a logarithmic factor $\log^{1/2}(2+t)$, while nonzero modes persist with exponential decay. The results close a long-standing gap for stability in Eulerian coordinates and provide methodological tools, including a Poincaré-type inequality and cancellations, that could extend to related viscous conservation laws.
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 Poincaré type inequality and a key cancellation.