# zbMATH — the first resource for mathematics

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 $$\gamma>\frac 32$$.

##### MSC:
 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