Limiting Absorption Principles and Linear Inviscid Damping in the Euler-Boussinesq System in the Periodic Channel.

Area of Science:

• Fluid Dynamics and Hydrodynamics
• Mathematical Physics
• Partial Differential Equations

Background:

• The long-time behavior of fluid flows is critical for understanding phenomena like turbulence and stability.
• Stratified flows, common in geophysical and astrophysical contexts, exhibit complex dynamics influenced by density gradients.
• The Euler equations, under the Boussinesq approximation, model such stratified flows in a simplified yet relevant manner.

Purpose of the Study:

• To analyze the long-time behavior of solutions to the 2D non-homogeneous Euler equations with Boussinesq approximation.
• To investigate the linearized system near a linearly stratified Couette flow.
• To prove inviscid damping for density and velocity fields for all positive Richardson numbers.

Main Methods:

• Analysis of the linearized system near a stratified Couette flow.
• Application of a limiting absorption principle to study time-decay properties of oscillatory integrals.
• Detailed asymptotic expansion of generalized eigenfunctions near the critical layer.

Main Results:

• Proof of inviscid damping for perturbed density and velocity fields with optimal rates for any positive Richardson number.
• Precise description of the spectrum of the linearized operator.
• Identification of essential spectrum and discrete neutral eigenvalues for large Richardson numbers.

Conclusions:

• Inviscid damping is a robust phenomenon in stratified 2D channel flows, occurring for all stratifications (positive Richardson number).
• The spectral analysis reveals a rich structure including continuous spectrum and discrete eigenvalues, explaining oscillatory modes.
• The findings contribute to a deeper understanding of stability and long-time dynamics in stratified fluid systems.