Vol. 13, No. 2, 2020

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On solvability and ill-posedness of the compressible Euler system subject to stochastic forces

Dominic Breit, Eduard Feireisl and Martina Hofmanová

Vol. 13 (2020), No. 2, 371–402
Abstract

We consider the barotropic Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of weak (distributional) solutions. Specifically, we find a sequence ${\tau }_{M}\to \infty$ of positive stopping times for which the Euler system admits infinitely many solutions originating from the same initial data. The solutions are weak in the PDE sense but strong in the probabilistic sense, meaning, they are defined on an a priori given stochastic basis and adapted to the driving stochastic process.

Keywords
stochastic compressible Euler system, weak solution, compressible Euler system, stochastic forcing, ill-posedness, convex integration
Mathematical Subject Classification 2010
Primary: 35Q31
Secondary: 35D30, 60H15