Abstract
The goal of this note is to show that, in a bounded domain \({\Omega \subset \mathbb{R}^n}\), with \({\partial \Omega\in C^2}\), any weak solution \({(u(x,t),p(x,t))}\), of the Euler equations of ideal incompressible fluid in \({\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t}\), with the impermeability boundary condition \({u\cdot \vec n =0}\) on \({\partial\Omega\times(0,T)}\), is of constant energy on the interval (0,T), provided the velocity field \({u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega}))}\), with \({\alpha > \frac13.}\)
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Bardos, C., Titi, E.S. Onsager’s Conjecture for the Incompressible Euler Equations in Bounded Domains. Arch Rational Mech Anal 228, 197–207 (2018). https://doi.org/10.1007/s00205-017-1189-x
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DOI: https://doi.org/10.1007/s00205-017-1189-x