Onsager's Conjecture with Physical Boundaries and an Application to the Vanishing Viscosity Limit

TL;DR

This paper extends Onsager's conjecture to bounded domains by relaxing boundary regularity conditions, which is significant for understanding boundary layer effects in the vanishing viscosity limit of fluid flows.

Contribution

It proves Onsager's conjecture under weaker boundary regularity assumptions, allowing for interior regularity and boundary flux continuity, relevant for boundary layer analysis.

Findings

Energy conservation holds with interior Hölder regularity and boundary flux continuity.

The new conditions align with boundary layer formation in vanishing viscosity scenarios.

Results support the analysis of boundary effects in fluid dynamics.

Abstract

We consider the incompressible Euler equations in a bounded domain in three space dimensions. Recently, the first two authors proved Onsager's conjecture for bounded domains, i.e., that the energy of a solution to these equations is conserved provided the solution is H\"older continuous with exponent greater than 1/3, uniformly up to the boundary. In this contribution we relax this assumption, requiring only interior H\"older regularity and continuity of the normal component of the energy flux near the boundary. The significance of this improvement is given by the fact that our new condition is consistent with the possible formation of a Prandtl-type boundary layer in the vanishing viscosity limit.