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Onsager’s Conjecture for the Incompressible Euler Equations in Bounded Domains


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.

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