The Stability and Decay for the 2D Incompressible Euler-Like Equations

This paper is concerned with the two-dimensional incompressible Euler-like equations. More precisely, we consider the system with only damping in the vertical component equation. When the domain is the whole space \(\mathbb {R}^2\), it is well known that solutions of the incompressible Euler equations can grow rapidly in time while solutions of the Euler equations with full damping are stable. As the intermediate case of the two equations, the global well-posedness and the stability in \(\mathbb {R}^2\) remain the outstanding open problem. Our attentions here focus on the domain \(\Omega =\mathbb {T}\times \mathbb {R}\) with \(\mathbb {T}\) being 1D periodic box. Compared with \(\mathbb {R}^2\), the domain \(\Omega \) allows us to separate the physical quantity f into its horizontal average \(\overline{f}\) and the corresponding oscillation \(\widetilde{f}\). By deriving the strong Poincaré inequality and two anisotropic inequalities related to \(\widetilde{f}\), we are able to employ the time-weighted energy estimate to establish the stability of the solution and the precise large-time behavior of the system provided that the initial data is small and satisfies the reflection symmetry condition.