Partial regularity of suitable weak solutions of the navier‐stokes equations

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

: The initial value problem for the nonstationary Navier-Stokes equation is considered. New results were obtained with the aid of various methods from modern functional analysis. The existence of unique and global (in time) solutions for 2-dimensional flows as well as the existence of unique and local (in time) solutions for 3-dimensional flows was established. In most of these works the word solution was interpreted in a generalized sense. The purpose is to deduce such existence theorems for 3-dimensional flows through a Hilbert space approach, making use of the theory of fractional powers of operators and the theory of semi-groups of operators. (Author)

Eberhard Hopf

Papers

Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Erhard Schmidt zu seinem 75. Geburtstag gewidmet