Global existence of martingale solutions to the three-dimensional stochastic compressible Navier-Stokes equations. (English) Zbl 1374.60119

The authors present conditions on the external force \(g\) and the initial data \(\rho|_{t=0}\), \((\rho u)|_{t=0}\) that yield existence of a global martingale solution to an initial-boundary value problem for a 3-dimensional compressible Navier-Stokes equation \(\rho_t+\operatorname{div}\,(\rho u)=0\), \[ d(\rho u)+[\operatorname{div}\,(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}\,u+\nabla p]\,dt=\rho g(\rho,u)\,dW \] with a homogeneous Dirichlet boundary on a bounded domain \(D\subseteq\mathbb R^3\) with the \(C^{2+}\)-smooth boundary \(\partial D\). The equation is driven by a finite-dimensional Wiener process \(W\), the viscosity coefficients are assumed to satisfy \(\lambda+\frac 23\mu\geq 0\) and the pressure \(p\) is given by the formula \(p=a\rho^\gamma\) for \(a>0\) and \(\gamma>\frac 32\).


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
60G44 Martingales with continuous parameter