On Ill- and Well-Posedness of Dissipative Martingale Solutions to Stochastic 3D Euler Equations

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.