On the Euler$$+$$Prandtl Expansion for the Navier-Stokes Equations

Optimization with the time-dependent Navier-Stokes equations as constraints

In this paper, optimal distributed control of the time-dependent Navier-Stokes equations is considered. The control problem involves the minimization of a measure of the distance between the velocity field and a given target velocity field. A mixed numerical method involving a quasi-Newton algorithm, a novel calculation of the gradients and an inhomogeneous Navier-Stokes solver, to find the opt...

On the barotropic compressible Navier-Stokes equations

We consider barotropic compressible Navier-Stokes equations with density dependent viscosity coefficients that vanish on vacuum. We prove the stability of weak solutions in periodic domain Ω = T and in the whole space Ω = R , when N = 2 and N = 3. The pressure is given by p(ρ) = ρ and our result holds for any γ > 1. Note that our notion of weak solutions is not the usual one. In particular we r...

The Stationary Navier-Stokes Equations

Having considered the linear Stokes equations, we will now bring back the nonlinear term and consider the nonlinear version of the Stokes system. The solution of these equations can be viewed as the limit to which the solution of the full N-S equations tends as t tends to infinity. Of course no one knows at this point if the solutions of the N-S equations do tend to some limit as t tends to inf...

Preconditioners for the incompressible Navier Stokes equations

Computational Fluid Dynamics is frequently used nowadays to understand the flow in rivers, blood veins, around cars and planes, etc. This tool can also be used to make better cars and planes and to design dams and dikes to protect against flooding. In this talk we consider simulation with the incompressible Navier Stokes equations. After discretization by the Finite Element Method and lineariza...

Numerical Methods for the Navier-Stokes equations