On Ill‐ and Well‐Posedness of Dissipative Martingale Solutions to Stochastic 3D Euler Equations. Issue 11 (28th September 2021)

• Record Type: Journal Article
• Authors: Hofmanová, Martina; Zhu, Rongchan; Zhu, Xiangchan
• Is Part Of: Communications on pure and applied mathematics. Volume 75:Issue 11(2022)
• Page Start: 2446
• Page End: 2510
• Publication Date: 2021-09-28
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.22023
• Languages: English
• ISSNs: 0010-3640

Abstract

Abstract: 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.