In this note we extend a 2018 result of Bardos and Titi (Arch Ration Mech Anal 228(1):197–207, 2018) to a new class of functional spaces \(C^{0,\alpha }_{\lambda }(\bar{\Omega })\). It is shown that weak solutions \(\,u\,\) satisfy the energy equality provided that \(u\in L^3((0,T);C^{0,\alpha }_{\lambda }(\bar{\Omega }))\) with \(\alpha \ge \frac{1}{3}\) and \(\lambda >0\). The result is new for \(\,\alpha =\,\frac{1}{3}.\) Actually, a quite stronger result holds. For convenience we start by a similar extension of a 1994 result of Constantin and Titi (Commun Math Phys 165:207–209, 1994), in the space periodic case. The proofs follow step by step those of the above authors. For the readers convenience, and completeness, proofs are presented in a quite complete form.

中文翻译：

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Hölog空间中不可压缩Euler方程的Onsager猜想$$ C ^ {0，\ alpha} _ {\ lambda}（\ bar {\ Omega}）$$Cλ0，α（Ω¯）

在本说明中，我们将Bardos和Titi的2018年结果（Arch Ration Mech Anal 228（1）：197–207，2018）扩展到一类新的函数空间\（C ^ {0，\ alpha __ \\\ lambda} （\ bar {\ Omega}）\）。结果表明，弱解\（\中，u \，\）满足规定，能量平等\（U \在L ^ 3（（0，T）; C 1-6 {0，\阿尔法} _ {\拉姆达}（ \ bar {\ Omega}}）\）和\（\ alpha \ ge \ frac {1} {3} \）和\（\ lambda> 0 \）。结果是\（\，\ alpha = \，\ frac {1} {3}。\）的新结果。实际上，结果要强得多。为方便起见，我们在空间周期情况下，以类似的方式扩展了Constantin和Titi的1994年结果（Commun Math Phys 165：207-209，1994）。这些证据将逐步遵循上述作者的证据。为了给读者带来方便和完整性，以非常完整的形式提供了证明。