Dissipative solutions and Markov selection to the complete stochastic Euler system

TL;DR

This paper introduces stochastic measure-valued solutions for the complete Euler system, establishing existence, weak-strong uniqueness, and Markov selection, advancing the mathematical understanding of stochastic fluid dynamics.

Contribution

It develops a novel framework for stochastic measure-valued solutions to the Euler system, including existence results and Markov selection, under energy balance conditions.

Findings

Existence of stochastic measure-valued solutions.

Weak-strong uniqueness principle established.

Existence of strong Markov selection for the martingale problem.

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 exists in the sense distributions). In particular, we show that: existence, weak-strong principle; a weak measure-valued solution coincides with a strong solution provided the later exists, all hold true provided they satisfy some form of energy balance. Finally, we show the existence of strong Markov selection to the associated martingale problem.