We are concerned with the question of well-posedness of stochastic, three-dimensional, incompressible Euler equations. In particular, we introduce a novel class of dissipative solutions and show that (i) existence; (ii) weak–strong uniqueness; (iii) nonuniqueness in law; (iv) existence of a strong Markov solution; (v) nonuniqueness of strong Markov solutions: all hold true within this class. Moreover, as a by-product of (iii) we obtain existence and nonuniqueness of probabilistically strong and analytically weak solutions defined up to a stopping time and satisfying an energy inequality. © 2021 Wiley Periodicals LLC.

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关于随机 3D 欧拉方程的耗散鞅解的不适定性和适定性

我们关注随机的、三维的、不可压缩的欧拉方程的适定性问题。特别是，我们介绍了一类新的耗散解决方案，并表明（i）存在；(ii) 弱-强唯一性；(iii) 法律上的非唯一性；(iv) 存在强马尔可夫解；(v) 强马尔可夫解的非唯一性：在这个类中都成立。此外，作为 (iii) 的副产品，我们获得了定义为停止时间并满足能量不等式的概率强解和解析弱解的存在性和非唯一性。© 2021 威利期刊有限责任公司。