Non-uniqueness of Admissible Solutions for the 2D Euler Equation with Vortex Data

Area of Science:

• Fluid dynamics
• Mathematical physics

Background:

• The 2D Euler equation describes inviscid fluid flow.
• Uniqueness of solutions is a fundamental question in fluid dynamics.
• The weak-strong uniqueness principle and Yudovich's uniqueness proof are key theoretical results.

Purpose of the Study:

• To investigate the sharpness of uniqueness principles for the 2D Euler equation.
• To demonstrate the existence of multiple bounded admissible solutions for specific initial conditions.
• To explore energy dissipation rates and their relation to the Onsager critical exponent.

Main Methods:

• Construction of initial velocity fields with specific vorticity profiles (truncated power-law vortices).
• Utilizing a self-similar subsolution approach.
• Application of the convex integration method to generate multiple solutions.

Main Results:

• Existence of infinitely many bounded admissible solutions for any $$.
• Demonstration of the sharpness of the weak-strong uniqueness principle.
• Extension of results for $$ and analysis of energy dissipation vanishing at $$.

Conclusions:

• The study highlights limitations in the uniqueness of solutions for the 2D Euler equation.
• The findings underscore the critical role of the exponent $$ in determining energy dissipation.
• Results provide insights into the Onsager conjecture and vorticity control in 2D fluids.