The stochastic 3D globally modified Navier-Stokes equations: existence, uniqueness and asymptotic behavior
The main results of the paper under review are related to the stochastic system of globally modified Navier-Stokes equations \[ \begin{cases} du(t)=[\nu\Delta u -F_N(\Vert u\Vert)[(u.\nabla) u]+\nabla p-F(u)]dt +\sum_{k=1}^\infty \sigma_k(u(t))d W_k(t) \\ \nabla.u=0\text{ in }(0,T)\times\mathcal{M} \\ u=0\text{ on }(0,T)\times\Gamma \\ u(0,x)=u_0(x),\ x\in\mathcal{M} \end{cases} \] where \((W_k)_{k\ge 1}\) is a sequence of independent one-dimensional standard Brownian motions, \(F_N(r)=\min\{1,N/r\}\), and \(\mathcal{M}\subseteq\mathbb{R}^3\) is a bounded domain with smooth boundary. For the above system, under suitable conditions, one proves the existence of a unique strong solution. One also studies the convergence of the strong solution for \(N\to\infty\), and one then proves proves the existence of a martingale solution.
globally modfied Navier-Stokes equations
scientific article; zbMATH DE number 6923959
scientific article