The goal of this note is to show that, in a bounded domain Ω ⊂ Rn, with ∂Ω ∈ C2, any weak solution (u(x, t) , p(x, t)) , of the Euler equations of ideal incompressible fluid in Ω × (0 , T) ⊂ Rn× Rt, with the impermeability boundary condition u· n→ = 0 on ∂Ω × (0 , T) , is of constant energy on the interval (0,T), provided the velocity field u∈ L3((0 , T) ; C0,α(Ω ¯)) , with α>13.

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 Hölder continuous with exponent greater than 1/3, uniformly up to the boundary. In this contribution we relax this assumption, requiring only interior Hölder 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.

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the “universality” of the Onsager exponent, α> 1 / 3 , concerning the regularity of the solutions, say in C ,α , that guarantees the conservation of the generalized entropy, regardless of the structure of the genuine nonlinearity in the underlying system.