Remarks on the Navier-Stokes Equations

Abstract

Necessary and sufficient conditions for the absence of singularities in solutions of the three dimensional Navier-Stokes equations are recalled. New global weak solutions are constructed. They enjoy the properties that the spatial integral of the vorticity magnitude is a priori bounded in time and that the space and time integral of the \(\frac{4}{3+\varepsilon}\) power of the magnitude of the gradient of the vorticity is a priori bounded. Vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. The two dimensional Hausdorff measure of level sets of the vorticity magnitude is studied and a priori bounds on an average such measure, are obtained. When expressed in terms of the Reynolds number and the Kolmogorov dissipation length η, these bounds are $$ \leq \frac{L^3}{\eta}(1+ Re^{-\frac{1}{2}})^{\frac{1}{2}}$$ The area of level sets of scalara and in particular isotherms in Rayleigh-Benard convection is studied. A quantity, r,t (x 0), describing an average value of the area of a portion of a level set contained in a small ball of radius r about the point x 0 is bounded from above by $$ _{r,t}(x_0)\leq C \kappa^{-\frac{1}{2}}r^{\frac{5}{2}} ^{\frac{1}{2}}$$ where κ is the diffusivity constant and is the average maximal velocity. The inequality is valid for r larger than the local length scale \(\lambda = \kappa ^{-1}\). It is found that this local scale is a microscale, if the statistics of the velocity field are homogeneous. This suggests that 2.5 is a lower bound for the fractal dimension of an (ensemble) average interface in homogeneous turbulent flow, a fact which agrees with the experimental lower bound of 2.35.