Dissipative solutions and Markov selection to the complete stochastic Euler system

Abstract

We introduce the concept of stochastic measure-valued solutions to the complete Euler system describing the motion of a compressible inviscid fluid subject to stochastic forcing, where the nonlinear terms are described by defect measures. These solutions are weak in the probabilistic sense (probability space is not a given ‘priori’, but part of the solution) and analytical sense (derivatives only exist in the sense distributions). In particular, we show that existence and weak-strong principle (i.e. a weak measure-valued solution coincides with a strong solution provided the later exists) hold true provided they satisfy some form of energy balance. Finally, we show the existence of Markov selection to the associated martingale problem.