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 τM 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
Milestones
Received: 24 May 2017
Revised: 8 December 2018
Accepted: 23 February 2019
Published: 19 March 2020
Authors
Dominic Breit
Department of Mathematics
Heriot-Watt University
Edinburgh
United Kingdom
Eduard Feireisl
Institute of Mathematics
Academy of Sciences of the Czech Republic
Prague
Czech Republic
Martina Hofmanová
Technical University Berlin
Institute of Mathematics
Berlin
Germany
Fakultät für Mathematik
Universität Bielefeld
Bielefeld
Germany