Non reflecting boundary conditions on artificial frontiers of the domain are proposed for both incompressible and compressible Navier-Stokes equations. For incompressible flows, the boundary conditions lead to a well-posed problem, convey properly the vortices without any reflections on the artificial limits and allow to compute turbulent flows at high Reynolds numbers. For compressible flows, the boundary conditions convey properly the vortices without any reflections on the artificial limits...

This paper is concerned with the two-species chemotaxis-Navier–Stokes system with Lotka–Volterra competitive kinetics $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\left(1\right)}_t+u·1=𝔻1-{\chi }_1·\left(1c\right)+{\mu }_{1}1(1-1-a_{1}2)\hfill & \text{in}\times (0,\infty ),\hfill \\ {\left(2\right)}_t+u·2=𝔻2-{\chi }_2·\left(2c\right)+{\mu }_{2}2(1-a_{2}1-2)\hfill & \text{in}\times (0,\infty ),\hfill \\ c_t+u·c=𝔻c-(\alpha 1+\beta 2)c\hfill & \text{in}\times (0,\infty ),\hfill \\ u_t+(u·)u=𝔻u+\nabla P+(\gamma 1+2)\Phi ,·u=0\hfill & \text{in}\times (0,\infty )\hfill \end{array}\right.\end{array}$$ under homogeneous Neumann boundary conditions and initial conditions, where is a bounded domain in R3 with smooth boundary. Recently, in the 2-dimensional setting, global existence and stabilization of classical solutions to the above system were first established. However, the 3-dimensional case has not been studied: Because of difficulties in the Navier–Stokes system, we can...

Ω being a bounded open set in R∙, with regular boundary, we associate with Navier-Stokes equation in Ω where the velocity is null on ∂Ω, a non-linear branching process (Yt, t ≥ 0). More precisely: Eω0(〈h,Yt〉) = 〈ω,h〉, for any test function h, where ω = rot u, u denotes the velocity solution of Navier-Stokes equation. The support of the random measure Yt increases or decreases in one unit when the underlying process hits ∂Ω; this stochastic phenomenon corresponds to the creation-annihilation of vortex...