Local strong solutions to the stochastic compressible Navier-Stokes system

• Published in: Communications in partial differential equations Vol. 43; no. 2; pp. 313 - 345
• Main Authors: Breit, Dominic, Feireisl, Eduard, Hofmanová, Martina
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01.02.2018; Taylor & Francis Ltd
• Subjects: Compressibility; Compressible fluids; Fluid dynamics; Fluid flow; local strong solutions; Manufacturing cells; Navier-Stokes equations; Navier-Stokes system; Statistical analysis; stochastic forcing; Stokes law (fluid mechanics); Viscous fluids
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2018.1442476

We study the Navier-Stokes system describing the motion of a compressible viscous fluid driven by a nonlinear multiplicative stochastic force. We establish local in time existence (up to a positive stopping time) of a unique solution, which is strong in both PDE and probabilistic sense. Our approach relies on rewriting the problem as a symmetric hyperbolic system augmented by partial diffusion, which is solved via a suitable approximation procedure. We use the stochastic compactness method and the Yamada-Watanabe type argument based on the Gyöngy-Krylov characterization of convergence in probability. This leads to the existence of a strong (in the PDE sense) pathwise solution. Finally, we use various stopping time arguments to establish the local existence of a unique strong solution to the original problem.