A KAM Approach to the Inviscid Limit for the 2D Navier-Stokes Equations

Area of Science:

• Fluid Dynamics
• Partial Differential Equations
• Mathematical Physics

Background:

• The inviscid limit of the Navier-Stokes equations is a fundamental problem in fluid dynamics.
• Understanding this limit is crucial for connecting viscous fluid behavior to inviscid models like the Euler equations.
• Previous research has faced challenges in achieving uniform-in-time results for quasi-periodic solutions.

Purpose of the Study:

• To investigate the inviscid limit for time-quasi-periodic solutions of the incompressible Navier-Stokes equations.
• To construct solutions that exhibit vanishing viscosity uniformly in time.
• To establish the first KAM (Kolmogorov–Arnold–Moser) result in the context of singular limit problems.

Main Methods:

• Construction of an approximate solution with a small error term.
• Application of a fixed-point argument using the constructed approximate solution.
• Analysis of the linearized Navier-Stokes operator's invertibility under specific conditions.

Main Results:

• Successfully constructed time-quasi-periodic solutions for the forced Navier-Stokes equation.
• Demonstrated that these solutions converge to solutions of the incompressible Euler equations as viscosity approaches zero.
• Achieved uniform-in-time convergence, independent of the external force's magnitude.

Conclusions:

• This work provides the first global and uniform-in-time positive result for the inviscid limit problem.
• It represents a novel application of KAM theory to singular limit problems in fluid dynamics.
• The findings offer new insights into the behavior of fluids at vanishing viscosity levels.