Very weak solutions for the stationary Stokes equations

Lagrangian Approximations and Weak Solutions of the Navier-Stokes Equations

The motion of a viscous incompressible fluid flow in bounded domains with a smooth boundary can be described by the nonlinear Navier-Stokes equations. This description corresponds to the so-called Eulerian approach. We develop a new approximation method for the Navier-Stokes equations in both the stationary and the non-stationary case by a suitable coupling of the Eulerian and the Lagrangian re...

Weak Solutions to the Stationary Incompressible Euler Equations

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...

On the Uniqueness of Weak Solutions of Navier-Stokes Equations: Remarks on a Clay Institute Prize Problem Uniqueness of Weak Solutions of Navier-Stokes Equations

We consider the Clay Institute Prize Problem asking for a mathematical analytical proof of existence, smoothness and uniqueness (or a converse) of solutions to the incompressible Navier-Stokes equations. We argue that the present formulation of the Prize Problem asking for a strong solution is not reasonable in the case of turbulent flow always occuring for higher Reynolds numbers, and we propo...

Conditions for the Local Regularity of Weak Solutions of the Navier-stokes Equations near the Boundary

In this paper we present conditions for the local regularity of weak solutions of the Navier-Stokes equations near the smooth boundary.