On Ill‐ and <scp>Well‐Posedness</scp> of Dissipative Martingale Solutions to Stochastic <scp>3D</scp> Euler Equations
Martina Hofmanová, Rongchan Zhu, Xiangchan Zhu
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.
Topics & Concepts
UniquenessMathematicsDissipative systemMartingale (probability theory)Markov chainStopping timeApplied mathematicsEuler equationsCompressibilityMathematical analysisPhysicsThermodynamicsQuantum mechanicsStatisticsNavier-Stokes equation solutionsGas Dynamics and Kinetic TheoryStochastic processes and financial applications